ActMAD: Activation Matching to Align Distributions for Test-Time-Training

Test-Time-Training (TTT) is an approach to cope with out-of-distribution (OOD) data by adapting a trained model to distribution shifts occurring at test-time. We propose to perform this adaptation via Activation Matching (ActMAD): We analyze activations of the model and align activation statistics of the OOD test data to those of the training data. In contrast to existing methods, which model the distribution of entire channels in the ultimate layer of the feature extractor, we model the distribution of each feature in multiple layers across the network. This results in a more fine-grained supervision and makes ActMAD attain state of the art performance on CIFAR-100C and Imagenet-C. ActMAD is also architecture- and task-agnostic, which lets us go beyond image classification, and score 15.4% improvement over previous approaches when evaluating a KITTI-trained object detector on KITTI-Fog. Our experiments highlight that ActMAD can be applied to online adaptation in realistic scenarios, requiring little data to attain its full performance.

Introduction
When evaluated in laboratory conditions, deep networks outperform humans in numerous visual recognition tasks [7][8][9]. But this impressive performance is predicated on the assumption that the test and training data come from the same distribution. In many practical scenarios, satisfying this requirement is either impossible or impractical. For example, a perception system of an autonomous vehicle can be exposed not only to fog, snow, and rain, but also to rare conditions including smoke, sand storms, or substances distributed on the road in consequence of a traffic accident. Unfortunately, distribution shifts between the training and Correspondence: muhammad.mirza@icg.tugraz.at  Figure 1. ActMAD is well suited for online test-time adaptation irrespective of network architecture and task. Our intended application is object detection on board of a vehicle driving in dynamically changing and unpredictable weather conditions. Here, we report the class-averaged Mean Average Precision (mAP@50) over adaptation time 1 , attained by continuously adapting a KITTI [12]-trained YOLOv3 [33] detector. Clear refers to the weather condition of the original dataset, mostly acquired in sunny weather, while Fog, Rain, and Snow are generated by perturbing the original images [15,19]. Gray vertical lines mark transitions between weather conditions. test data can incur a significant performance penalty [2,23,28,31,38,40]. To address this shortcoming, test-time-training (TTT) methods attempt to adapt a deep network to the actual distribution of the test data, at test-time.
Existing TTT techniques limit the performance drop provoked by the shift in test distribution, but the goal of online, data-efficient, source-free, and task-agnostic adaptation remains elusive: the methods that update network parameters by self-supervision on test data [10,28,36] can 1 Time (seconds) for adaptation is hardware specific. For reference, we run these experiments on an RTX-3090 NVIDIA graphics card and it takes ∼75s to adapt to 3741 images (one weather condition) in the KITTI test set. only be used if the network was first trained in a multitask setup, the approaches that minimise the entropy of test predictions [26,38], or use pseudo-labels, or pseudoprototypes [22,23], are not easily adapted to object detection or regression, and the ones that update statistics of the batch normalization layers [31,34] are limited to architectures that contain them. As a consequence of these limitations, while existing methods fare well in image classification, none of them is well suited for the scenario where the need for online adaptation is the most acute, that is, object detection on board a car driving in changing weather conditions.
Our goal is to lift the limitations of the current methods and propose a truly versatile, task-, and architectureagnostic technique, that would extend TTT beyond image classification and enable its deployment in object detection for automotive applications. To that end, we revisit feature alignment, the classical domain adaptation technique [35,43], also adopted by several TTT algorithms [23,28,31,34]. It consists in aligning the distribution of test set features to that of the training set. By contrast to previous methods, that align the distributions of entire channels in the ultimate layer of the feature extractor, ActMAD brings the feature alignment approach to another level by individually aligning the distribution of each feature in multiple feature maps across the network. On the one hand, this makes the alignment location-aware. For example, the features from the bottom of the image, most often representing road and other vehicles, are not mixed with features from the top part, more likely to describe trees, buildings, or the sky. On the other hand, it results in a more fine-grained supervision of the adapted network. Our ablation studies show that ActMAD owes most of its performance to these two contributions. Additionally, while several authors suggested aligning higherorder moments [23,43], we demonstrate that aligning means and variances should be preferred when working with small batches. Unlike methods that only update mean and variance [31,34], or affine parameters of batch normalization layers [23,31,34], ActMAD updates all network parameters. Our approach is architecture-and task-agnostic, and does not require access to the training data or training labels, which is important in privacy-sensitive applications. It does require the knowledge of activation statistics of the training set, but this requirement can be easily satisfied by collecting the data during training, or by computing them on unlabelled data without distribution shift.
Our contribution consists in a new method to use Activation Matching to Align Distributions for TTT, which we abbreviate as ActMAD. Its main technical novelty is that we model the distribution of each point in the feature map, across multiple feature maps in the network. While most previous works focus on combining different approaches to test-time adaptation, ActMAD is solely based on feature alignment. This is necessary for our method to be applicable across different architectures and tasks, but we show that discarding the bells and whistles, while aligning activation distributions in a location-aware manner results in matching or surpassing state-of-the-art performance on a number of TTT benchmarks. Figure. 1 presents the performance of ActMAD in a simulated online adaptation scenario, in which an object detector runs onboard a vehicle driven in changing weather conditions. Most existing methods cannot be used in this setup, because they cannot run online, or because they cannot be used for object detection. Note, that our method recovers, and even exceeds, the performance of the initial network once the weather cycle goes back to the conditions in which the detector was trained.

Related Work
Unsupervised domain adaptation. The idea of aligning neural feature distributions to bridge the domain gap between training and test data was first used for unsupervised domain adaptation (UDA). In CORAL [35], the alignment is performed by first whitening the target domain features and then re-coloring them with covariances computed on the source domain. Alternative methods train the network to minimize a difference of features statistics, like the higherorder central moments [43], or the higher-order cumulant tensor [3]. These UDA algorithms cannot be used for testtime-training (TTT), because they work offline and require access to the source training data and entire test data. Our ActMAD, and some existing TTT approaches e.g. [28,31], also explicitly compute and align feature statistics of the training and test data, but perform online adaptation without access to the training data.
Test-Time-Training. TTT consists in adapting a network to a distribution shift between the test and training data at test-time without access to the training data. The technique that gave the name to the whole group of methods [36] augments the training routine with a self-supervised task, which enables re-training the network on the test set even though no test labels are available. The original work employed rotation prediction [13] as the auxiliary task, but subsequent works [10,28] replaced it with the Masked Autoencoder reconstruction task [16] or contrastive learning [4]. These methods improve performance, but cannot be used if the network was not trained with the auxiliary task. Moreover, contrastive learning requires a large portion of unlabelled data to be effective. By contrast, our ActMAD can be used to adapt a network trained with an arbitrary protocol, and requires very little data to attain its full performance.
Instead of requiring that the network is trained in a multitask setup, CoTTA [39] employs a student-teacher setup only during test-time adaptation. ActMAD performs on par with CoTTA on its own benchmark, while consuming less memory and computation which CoTTA needs to run the teacher ReLU ReLU Figure 2. Schematic of ActMAD. Given a pre-trained model and statistics of the clean activations from the training data, it aligns the activation responses from the shifted test data to the clean activations at test-time. We model the activation distributions in terms of the means and variances of each activation, such that the statistics have the same shape as the feature maps. The statistics of the training activations are pre-computed on the training set, or computed on unlabelled data without distribution shift.
network. CoTTA relies on the entropy of the predicted class distribution to transfer the knowledge from the teacher to the student, so it cannot be used for detection-specific architectures that regress the bounding box coordinates. By contrast, ActMAD is seamlessly embedded in detection systems. While other methods adapt network parameters, T3A [22] casts test-time adaptation as prototype learning. It relies on pseudo-labels to create test-set-specific pseudo-prototypes, which then substitute the classifier learned on the training set. ActMAD outperforms T3A in classifying ImageNet-C and CIFAR-100C images.
Similar to the authors of the T3A, Boudiaf et al. [2] propose to forego updating the network parameters and develop an alternative prediction strategy. It generates class likelihoods by minimizing an objective function that combines a term promoting consistency of the predictions with network outputs and Laplacian regularization. Their method is effective for alleviating the shift in the prior distribution, but brings little improvement when the shift affects appearance of test images, which is the focus of our work.
Another group of TTT methods adjusts network parameters using entropy-based objectives on the test data. SHOT [26] minimizes the expected entropy of individual predictions, while maximizing the entropy of the predicted classes over the entire test set. TENT [38] adjusts the scale and shift parameters of batch normalization layers to minimize the entropy of test predictions. The entropy-based loss functions prevent direct application of these methods to object detection and regression, but ActMAD outperforms them by a fair margin also in image classification.
ActMAD is most closely related to TTT approaches that align statistics of training and test features. Along these lines, NORM [34] and DUA [31] update the batch normalization statistics computed on the training set to match the distribution of the test set. CFA [23] goes further and updates the affine layers in the network to match class-conditioned higher-order central moments of the last-layer test features to those computed for the training set. The reliance on classconditioning prevents application of CFA to object detection. By contrast, ActMAD is free from this constraint. It aligns means and variances of individual features in multiple layers across the network, and updates the entire parameter set. This makes it outperform DUA and NORM, and match the performance of CFA in image classification.

ActMAD
We are given a deep network f (x; θ), where x is an input image and θ is a vector of parameters. We are also given a parameter vector θ * , obtained by training the network on a dataset S of images and their annotations. Our goal is to use f to process a test set T of images that may differ in appearance from the ones in S, but are consistent with them in terms of their content. In many applications, the test images are delivered in a stream, and while we can compose them into batches of moderate size, we are not capable of iterating over the test set. In our flagship application, f is an object detector deployed in a car, S is a sequence of images acquired on a sunny day, and T is a video stream acquired during driving the car in changing weather conditions. To address both the detection and classification scenarios, we impose no constraints on the form of f (x; θ). To ensure maximum versatility of ActMAD, we make no assumptions about the architecture of the network, or the training process.
The requirements of task-and architecture-independence rule out the use of techniques that rely on the output of the network to be a probability distribution [22,26,38,39], or employ auxiliary tasks during training [28,36]. We thus follow the only viable design choice: we treat selected layers of the network as random variables and align the distribution of these variables in the test set to that of the training set. An overview of our method is presented in Figure. 2.
Location-aware activation alignment. We denote the activation responses of the l-th layer of the network f computed for image x by a l (x; θ). This is a feature map of shape C × W × H, denoting the number of channels and its spatial dimensions. Typically, a l does not depend on all model parameters θ, but we abuse the notation in the interest of simplicity. Existing TTT methods based on feature alignment [23,28,31,34] treat all the features in a channel as instantiations of the same random variable, irrespective of their location in the image. By contrast, we assume that different locations in the feature map may be distributed differently. This is particularly the case in photographs, because humans frame objects of interest when taking pictures, and in driving scenarios, where different objects occur in different regions of the image. We therefore separately align each of the C × W × H activations in the l-th layer. This results in a stronger supervisory signal than the classical approach of integrating over the spatial dimensions and aligning the distributions of C channels at the output of the encoder. Our experiments show that this stronger supervision leads to faster and more effective adaptation.
Aligning means and variances. To align activations, we could use one of the existing feature alignment methods, for example adversarial alignment [11], but this would require access to the training data, which we are not granted. We are therefore limited to methods that characterize the distribution in terms of statistics, which could be pre-computed on the training data, and enforce distribution alignment by minimizing the discrepancy of the test statistics with those of the training set. The classical way to do this is to minimize the difference between higher-order sample moments of the distributions [3,43]. However, unlike in domain adaptation, where moment-based methods gained popularity, we need to adapt the distributions based on small samples, since we model each location in each channel separately, and run the adaptation online, on small batches. We thus forego higher-order statistics and use feature means and variances where N D is the number of images in D, i.e. the dataset over which the estimates are computed. For each selected layer l, we pre-compute the activation statistics of the training set S, During adaptation, we compute the same statistics for each batch B of the test set, and minimize Aligning multiple layers. We found that aligning just the statistics of the final layer, following the common practice [23,28], is not enough for fast and effective distribution alignment. We show this in an ablation study in Section 4.5. We therefore define a set L of multiple layers for alignment and the corresponding alignment objective as Intuitively, the aligned layers should be distributed evenly across the network. In the case of convolutional architectures with the standard convolution-normalization-ReLU sequences, we found that it is best to align the features obtained after normalization and before the nonlinearities.
Updating the complete parameter set. Finally, while numerous existing methods only update affine parameters or statistics of the batch normalization layers [23,31,34,38], we update all the parameters of the adapted network. It might seem a priori that this increases the risk of making the parameter vector drift away from its initial value, but continuously aligning a very rich set of statistics actually allows the network to recover its initial performance after the distribution of the test data becomes consistent with that of the training set, as shown in Figure. 1. The ablation studies in Section 4.5 further confirm that there is no significant performance difference between the two strategies, and thus, we opt for the more deployment-friendly of the two alternatives and make a gradient step on for each batch B of the test data T .   Table 4. Continuous Adaptation (following CoTTA [39]) to distribution shifts at test-time. We report the mean error of 10 runs, while shuffling the order of the corruptions for each run.

Experimental Evaluation
We first summarize the datasets and baselines used for the evaluation, and then discuss our results.

Datasets
We ran experiments on the following data sets: -CIFAR-10C and CIFAR-100C [19] were created by corrupting test images of the CIFAR-10 and CIFAR-100 datasets [24]. We used them to benchmark the performance of ActMAD in image classification, and for ablation studies. Following the common protocol [28,31,34,36,38], we used 15 corruption types and 5 corruption levels.
-ImageNet-C [19] contains ImageNet [5] test images, corrupted in the same way as those of CIFAR-10/100C. We used them for large-scale classification experiments. Again, we tested for 15 corruption types and 5 severity levels.
-We used the KITTI dataset [12] to evaluate the per- formance of ActMAD on object detection for automotive applications. KITTI contains 8 classes of traffic participants, and was recorded mostly in clear weather. To simulate degrading weather conditions on the KITTI dataset, we used physics-based rendering of fog and rain [15], and followed Mirza et al. [30] to simulate snow. For testing, we used the dedicated test split of each dataset. On KITTI, however, the annotations for the test set are not publicly available, so we divided the public train set randomly into train and test split with equal proportion. Here, we report the results for the highest distribution shift severity. Results for lower severity are provided in the supplemental.

Baselines
We compared ActMAD against the following approaches: -Source is the model trained on source data and evaluated on test data without any adaptation. -TTT [36] adapts the network to test data using selfsupervision from a rotation prediction task. -TTT++ [28] employs a combination of self-supervised contrastive learning and feature alignment. -P-L is our adaptation of the pseudo-labeling approach [25]

Image Classification
To evaluate ActMAD on image classification, we adapted a Wide-ResNet-40-2 [42] (WRN-40-2) from the Robust Bench [19] and a Vision Transformer [6] (ViT-B/16) backbone on the CIFAR-C datasets. For Imagenet-C dataset, we used an ImageNet pre-trained ResNet-18 [17] from the PyTorch model zoo, following previous work [31]. Results on ImageNet-C with other backbones are provided in the supplemental. We used a batch size of 128 for all image classification experiments. ActMAD was run with the learning rate of 1e−2 for CIFAR-10C, 1e−3 for CIFAR-100C and 25e−5 for ImageNet-C, while the baselines used hyperparameters reported by their authors. Following prior literature [22,31,38], we performed the experiments online, using each test image to adapt the network only once. Since SHOT and TTT++ are better suited for the offline scenario, we additionally run them for 500 epochs on the entire test sets of the CIFAR-C data sets, with an early stopping criterion, according to the official TTT++ implementation 2 .
Results for CIFAR-10/100C for WRN-40-2 are reported in Table. 1 and 2, while the ViT results are listed in Table. 3. ActMAD outperforms other online methods in classifying CIFAR-10C and CIFAR-100C images by a fair margin, establishing a new state-of-the-art for both Transformer-based and convolutional architectures. When adapting the wide ResNet to CIFAR-10C, there is almost no gap between Act-MAD and the offline approaches, while on CIFAR-100C offline adaptation fares better. For the ViT, we also achieved a new state-of-the art on CIFAR-10C and CIFAR-100C, outperforming even CFA [23], designed specifically for robustifying vision transformers.
The results for the ImageNet-C dataset are shown in Table. 5. ActMAD comes in second after CFA [23], but the  Table 6. Mean Average Precision (mAP@50) for a KITTI pre-trained YOLOv3 tested on rain, fog and snow datasets. Higher is better. a) Results for the most severe fog level, i.e. only 30m visibility. b) Results for the most severe rain level, i.e. 200mm/hr rain intensity. c) Results for snow. Best mAP is shown in bold, while the second best is underlined.  performance gap in classification error is only 0.1 percentpoint. We suspect the advantage of CFA to stem from its alignment of class-conditioned statistics, which may become more important for the rich and diverse set of ImageNet classes than for the limited sets of classes of the CIFAR datasets. However, the use of class-conditional statistics limits CFA to the classification task.
Continuous adaptation to pertubations. Recently, Wang et al. evaluated their TTT method CoTTA [39] by continuously adapting to different corruptions, instead of adapting to only a single corruption at a time. We tested ActMAD and other baselines in such a scenario on CIFAR-10/100C. The results, listed in Table. 4, show that ActMAD performs on par with CoTTA, and outperforms other baselines, while consuming less memory and computing power than CoTTA, which relies on a student-teacher setup.

Object Detection
For object detection experiments, we adapted a KITTIpretrained YOLOv3 [33] to degrading weather conditions, such as fog, rain, and snow. We used a learning rate of 1e−4, and a batch size of 30. We compared our object detection results with DUA, NORM, and TTT. DUA and NORM could be applied to detector adaptation without any modifications, thanks to the fact that they do not rely on the network to produce normalized probabilities and are not limited to adapting models trained in a multi-task setting. By contrast, to deploy TTT, we had to re-train the detector with the auxiliary, self-supervised rotation prediction task. We describe the implementation details in the supplemental. The results, reported in Table. 6, show that ActMAD outperforms the baselines by a significant margin. Additionally, in Figure. 3 we plot the performance of the network during adaptation, showing that ActMAD stabilizes its precision as adaptation progresses, while the baselines provoke performance fluctuations. Needless to say, consistent level of performance is of utmost importance in safety-critical applications, like autonomous driving.
We also tested ActMAD in continuous online adaptation, where different weather conditions are encountered interchangeably, and show one such result in Figure. 1, while deferring two more plots to the supplemental. After adaptation to a whole weather cycle, ActMAD can quickly re-adapt to the conditions on which the network was initially trained, and even exceed its initial performance.

Ablation Studies
Effect of batch size. The number of samples used by Act-MAD to compute the location-aware statistics is equal to  the number of batch elements, and therefore much lower than the number of samples used to compute the channelwise statistics in competing methods. Statistics computed on small samples have higher variance, which could adversely affect adaptation performance. We investigate how decreasing the batch size affects the performance of ActMAD. Like in the image classification experiments, we use an ImageNet pre-trained ResNet-18 from the PyTorch model zoo. We decrease the learning rate in proportion to the decreasing batch size, following Goyal et al. [14]. As shown in Figure. 4, limiting the batch size decreases the performance gently, and even batches as small as 10 images still let us achieve 13.4 percent-point improvement over the un-adapted model.
Design alternatives. To verify how our design choices contribute to the performance of ActMAD, we removed each of its key features and evaluated the resulting algorithms on image classification and object detection. We adapted a wide ResNet from CIFAR-10 to CIFAR-10C, and YOLOv3 from KITTI to KITTI-Fog, in the same setup as in previous experiments. As can be seen in Table. 7, a large performance drop was provoked by limiting the alignment to the last feature map, as opposed to multiple feature maps throughout the network, and by aligning channel, instead of pixel, statistics.
This attests to the importance of fine-grained supervision, distributed over multiple layers and locations in the feature maps. Unsurprisingly, the use of location-aware statistics results in higher performance gain for KITTI (∼ 10 percentpoint) than for CIFAR (∼ 1 percent-point). This is expected, since road scenes of KITTI are much more structured than the CIFAR images. We also hypothesize that ActMAD's localized feature alignment benefits object detection more, because detection relies on local features to a higher degree than the location-agnostic image classification. The type of statistics used for feature alignment seems to make no difference for CIFAR-10C, but aligning means and variances gives better results than aligning higher-order central moments for the KITTI object detector. This difference might stem from the fact that the KITTI experiments use almost ten times smaller batch size, and that low batch size affects higherorder moment estimates more than estimates of means and variances. Finally, limiting the parameter update to channelwise affine transformations results in a small performance drop in object detection, while almost not affecting the image classification accuracy. However, since we observed no drift in the continuous adaptation experiment and limiting the update might require manually inserting affine layers to the network, we prefer to perform the full update.

Conclusion
We propose a new method -ActMAD -for online testtime-training based on location-aware feature alignment. In contrast to many previous TTT approaches, our ActMAD is task-and architecture-agnostic. It goes beyond image classification and can be readily applied to object detection. ActMAD outperforms existing approaches by a considerable margin, while being more stable and adapting faster to new distributions. The power of ActMAD stems from finegrained supervision which results from individually aligning each pixel in multiple feature maps across the network. Experiments show that the fine-grained supervision provided by ActMAD from location-aware feature alignment is especially helpful for tasks which require a holistic understanding of the scene, for example object detection.

Supplemental Material
In the following, we present additional continuous online adaptation scenarios in which we test ActMAD (Section A), provide all implementation details for baselines and our experiments to support reproducibility (Section B) and present additional ablation studies (Section C).

A. Continuous Online Adaptation
Our location-aware feature alignment is especially helpful for the task of object detection. The fine-grained supervision helps ActMAD to adapt to different weather scenarios in a dynamic and efficient manner. Moreover, ActMAD also shows promising results while adapting to different weather scenarios in a continuous manner. In the main manuscript (Figure 1), we show such a simulated continuous adaptation scenario (Clear→Fog→Snow→Rain→Clear). We further test ActMAD in similar scenarios where the weather conditions are encountered in a different sequence. In Figure 5a and 5b, we present results for adaptation with ActMAD in two such scenarios.
The results show that ActMAD can achieve impressive results in different scenarios and the sequence of weather conditions which are encountered does not matter. This is close to the requirements in the real-world, where different weather conditions can occur in any order during driving. We also see from the results that Act-MAD can even improve the performance of a KITTI [12] pre-trained YOLOv3 [33] on the validation split from the dataset (which represents the Clear weather condition). For example, in Figure 5b, where order of the weather conditions is Clear→Clear→Fog→Snow→Rain→Clear, Act-MAD achieves 1.0% higher Mean Average Precision while adapting from Clear→Clear at the beginning of the adaptation cycle. This is because even though the adaptation is performed from Clear→Clear, the activation statistics could be different between the training and inference, e.g., due to slight change in the environmental conditions or scenery. Our ActMAD helps to rectify this mismatch to obtain an increase in performance.

B. Implementation Details
Hardware: All our experiments are performed on a single NVIDIA ® GeForce ® RTX 3090 and a GPU server consisting of four Quadro RTX 8000.
ImageNet-C: All ImageNet-C experiments use ImageNet pre-trained models. For the main results with ResNet-18 [17] we use the pre-trained weights from the PyTorch [32] model zoo. For experiments with ResNet-50 using Group Normalization [41] (Section C.1), we use pre-trained weights from a third-party open source repository 3 . For experiments with a DeepAug [18] pre-trained ResNet-50 (Section C.2), we use the pre-trained weights from Robust Bench 4 . For experiments on ImageNet-C with ViT-B/16 model (Section C.4), we use the pre-trained weights from the open source timm library 5 .
KITTI: All KITTI [12] experiments start with a model pre-trained on the original KITTI dataset. As it is collected mostly in clear (i.e. sunny) weather conditions, we refer to it as KITTI-Clear. We adapt this model to degrading weather conditions, i.e. fog [15], rain [15] and snow [19]. For KITTI-Clear, we re-train an MS-COCO [27] pre-trained YOLOv3 in a supervised manner on the train split of the KITTI dataset. The model is trained for 100 epochs with a batch size of 30, while all the other training details are kept constant as in the PyTorch implementation of YOLOv3 6 .
Baselines: Since all the baselines used to compare our results are primarily tested for object recognition on CIFAR-10/100C [19] and ImageNet-C [19], we use their official open-source codes and use all the hyper-parameters, which they report in their paper.
For object detection experiments, only DUA [31] provides official results 7 . Therefore, we implement NORM [34] for object detection using their official repository 8 . Since NORM is a gradient-free approach, it does not rely on any hyper-parameters. We use a batch size of 30 to obtain all the results for NORM.
TTT [36] uses rotation prediction as an auxiliary task for test-time training. TTT is also primarily tested for object recognition by its authors. We implement TTT for object detection for the purpose of comparison in our paper. We use the same batch size (i.e. 30), as for ActMAD. For jointtraining with the auxiliary rotation prediction task we take the output from the feature encoder of YOLOv3 and use a single-layer linear classifier head to predict the rotation of the input image. The hyperparameters used for the KITTI pre-training (for ActMAD experiments) and joint-training are kept the same. However, we find that training a network jointly for the classification task of rotation prediction and object detection requires special design choices. For    example, we use a loss scaling factor of 0.4 for the selfsupervised task during joint-training to make the network converge. Similarly, the same loss scaling factor is used during test-time training. The evaluation protocol is kept consistent for all baselines and our method.
We also implemented TTT++ [28] on YOLOv3 in order to use it as a baseline for object detection experiments. We used the same prediction head, hyper-parameters and the augmentations (for creating two augmented views for SimCLR [4] training) used by the original TTT++ paper for their experiments on CIFAR-10/100. We experimented with several design choices for joint-training (object detection and contrastive SimCLR task), each for a total of 500 epochs on the KITTI train set, starting from the pre-trained weights on MS-COCO. However, despite the best of our efforts we could not make the joint-training converge. A potential issue could be the requirement of using larger batch sizes for training the contrastive learning task. Initially, we tested with a batch size of 100 for joint training by keeping the original aspect ratio of KITTI images (i.e. 370 × 1224). However, realizing that we might require larger batch sizes, we had to downscale the KITTI images equal to the aspect ratio of ImageNet images (i.e. 224 × 224), to be able to increase the batch size to 300, but in this case the regression loss for object detection did not converge and we get low performance on the validation set of KITTI. We suspect that it might be because of the drastic decrease in the image size, as compared to the original aspect ratio of images in the KITTI dataset, which is unsuitable for the object detection task on the KITTI dataset.

C. Additional Results
In this section we provide additional ablation studies, results on lower severity levels for ActMAD, detailed results for the vision transformer backbone [6] and the Continuous Adaptation experiment.

C.1. Different Normalization Techniques
Recent test-time training approaches which adapt the network statistics, e.g. NORM [38] and DUA [31], criti-  Table 9. Top-1 Classification Error (%) for all corruptions in CIFAR-100C (level 4 -1), highest severity is reported in the main manuscript. Lower is better. The results were obtained by adapting a WRN-40-2 backbone, trained on CIFAR-100, to CIFAR-100C. ActMAD results are averaged over 10 runs, and we report the mean error and its standard deviation.
cally rely on batch normalization [21]. However, if the network is equipped with other forms of normalization, e.g. Instance [37], Layer [1] or Group Normalization [41], these approaches are not applicable. In contrast, our ActMAD is agnostic to the architecture and can work with any type of normalization applied in the network. We already showed in the main manuscript ( Table 3) that it can be applied to a Transformer-based architecture, which uses Layer Norm. In order to test the performance of ActMAD with different normalization layers inside the convolution-based architectures, we test ActMAD with a ResNet-50 equipped with Group Normalization. Figure 6 shows the adaptation results of a ResNet-50 with both Batch Normalization (BN) and Group Normalization (GN). Using GN, our ActMAD performance decreases by only 1.8% over the same architecture using BN.

C.2. Robustifying an Already Robust Model
ActMAD can be effectively used with any off-the-shelf pre-trained model. To test this, we apply ActMAD for test-time adaptation of an already robust, AugMix [20] pretrained ResNet-50. We further decrease the batch size for adaptation on this backbone. Results for this experiment (highest severity) are provided in Figure 7. We find that Act-MAD decreases the error significantly even for an already  Table 10. Top-1 Classification Error (%) for all corruptions in ImageNet-C (level 4 -1), highest severity is reported in the main manuscript. Lower is better. The results were obtained by adapting a ResNet-18 backbone, trained on ImageNet, to ImageNet-C. ActMAD results are averaged over 10 runs, and we report the mean error and its standard deviation.  Table 11. Top-1 Classification Error (%) for all corruptions in CIFAR-10/100C and ImageNet-C (level 5). Lower is better. All results are obtained by using a ViT-B/16 backbone. ActMAD results are averaged over 10 runs, and we report the mean error and its standard deviation. robust model. For example, the Source (without adaptation) top-1 error (%) over the 15 corruptions on ImageNet-C is 61.1%, while after ActMAD adaptation the error decreases to merely 43.5%. Furthermore, we also see that while applying ActMAD to a robust model, the decrease in performance with decreasing batch size is also minimal (similar to the ablation study in the main manuscript, Figure 4, with ResNet-18 backbone). For example, while decreasing the batch size from 125 to 10, the error increases by only 4.1%.

C.3. Lower Severity Results
In the main manuscript we provide results for the highest severity (Level 5) in the Robust Bench [19]. Our ActMAD achieves impressive gains in performance even for lower severity levels. In Table 8 and Table 9 we provide the mean and standard deviation of results (over 10 random runs) ob-tained by ActMAD on lower severities (i.e. Levels 4 -1) on CIFAR-10/100C. We use the Wide-ResNet-40 model for these evaluations. Furthermore, in Table 10, we provide the mean and standard deviation of results over 10 random runs on the large scale ImageNet-C (with ResNet-18) dataset for lower severities (Levels 4 -1). The results show that ActMAD can even be applied when the distribution shift between the train and test data is not too large.

C.4. Vision Transformer
ActMAD can also be seamlessly applied to the Vision Transformer backbone. In the main manuscript we provide mean results over the 15 corruptions for CIFAR-10/100C (Level 5). For reference, in Table 11 we provide results for each individual corruption for the highest severity in the CIFAR-10/100C. Additionally, we also provide ActMAD re-  Table 12. Top-1 Classification Error (%) for all corruptions in CIFAR-10/100C (level 5), while continuously adapting to corruptions. Lower is better. These results were obtained by adapting a Wide-ResNet-40-2 backbone, trained on CIFAR-10/100, to CIFAR-10/100C. ActMAD results are averaged over 10 runs, and we report the mean error and its standard deviation. The lowest error is shown in bold, the second best is underlined.
sults for the highest severity on the ImageNet-C benchmark.

C.5. Continuous Adaptation to Perturbations
In the main manuscript (Table 4) we compare with CoTTA [39] on their continuous adaptation benchmark and provide the mean error over the 15 corruptions (highest severity) in the CIFAR-C benchmark. In Table 12 we provide detailed results, averaged over 10 random runs, for each individual corruption for documentation purposes.


Enhancing brush tyre model accuracy through friction measurements

Accurate tyre models are important to ensure valid and reliable simulation of vehicle behaviour. To this purpose, the Magic Formula (MF) became the de facto standard for vehicle dynamics simulations, despite requiring many empirically derived coefficients. This paper shows how the accuracy of a simple physical-based brush-type tyre model can be enhanced to simulate tyre behaviour that closely matches MF results. To do so, the real, highly complex rubber friction characteristics are incorporated into the brush model. The friction characteristics were obtained from friction measurements with a tread rubber block. The developed model is validated against experimental tyre data obtained on a flat-track test rig and the corresponding MF model. Results show that the inclusion of friction characteristics allow accurate simulation of longitudinal and lateral slip conditions over a wide range of normal loads with the simple brush model. At extreme loading scenarios, the simulation accuracy deteriorates because of the significant influence of highly nonlinear deformation behaviour of the tyre that is not accounted for in the brush model formulation.

Introduction
The successful development of virtual engineering capabilities of automotive manufacturers has led to a significant reduction in vehicle development times. For example, physical prototypes and the associated physical testing can be replaced by full-vehicle simulations provided that they are accurate and reliable. The quality of such simulations depends on the quality of the constituent models. With regard to tyre simulations, most tyre models currently used are derived from data obtained with indoor flat-track test rigs [1]. Once test data are collected, they are commonly parameterised with an empirical model, the Magic Formula (MF) [2].
In its most basic form, the MF is a simple equation containing nested trigonometric functions with several scaling coefficients. Over the years, the number of parameters was increased to enhance model accuracy, with the current formulation relying on over 100 coefficients. As a result, the creation of an accurate MF model requires a comprehensive experimental dataset. To reduce the considerable MF parameterisation effort, several lowparameter models have been developed, some of which build on the understanding of tyre physics [2,3]. One of the simplest models is the brush model, which provides a good physical description of the tyre-road interaction but lacks simulation accuracy. To improve this aspect, several extended brush model formulations have been proposed. For example, the inclusion of an arbitrary pressure distribution, as opposed to an n-order polynomial commonly seen in the literature, is considered in [4], allowing for a more realistic pressure distribution (e.g. one taken from experiments) to be included. [2] proposes a brush model with a flexible carcass to enhance the accuracy of the aligning moment. Thermal effects are formulated in [5], with the inclusion of Persson's rubber friction model and, thus, a dependence on surface roughness. In [6], the footprint geometry and lateral curvature of the tyre is considered in a 3D approach, and used to successfully predict longitudinal behaviour of a tyre on both plane and curved surfaces.
Although, the extended formulations improve simulation accuracy, they also increase model complexity. Combining several of such extensions to improve different simulation aspects can result in a highly complex model, which will again make parameterisation difficult. In addition, the computational burden will increase and the model will be less likely to be able to run in real-time simulations [5].
This paper presents a novel approach to improve the accuracy of a simple brush model, by considering the real tread-rubber/ground friction characteristics. The model formulation is deliberately kept as simple as possible to demonstrate the benefits and importance of accurately capturing the friction characteristics. Since the MF is a widely used tyre model (owing to its high utility and versatility), the brush model results are compared to the MF as a 'benchmark'. Note that there is no intention of replacing the MF. Rather, the comparison serves to show the utility of the integration of accurate friction measurements with a very simple brush model.
Rubber friction is a physical phenomenon that is not yet fully understood, and depends on many physical parameters, including temperature, sliding velocity and surface roughness [7]. Despite excellent work developing a robust, mathematical theory of rubber friction (e.g. [7,8]), comparisons of rubber friction theory to experiments show that our understanding is not complete (e.g. [9]). Thus, to be able to simulate the real friction characteristics of a tyre, friction measurements were carried out with a tread rubber block using a purpose-built test rig under highly controllable conditions. Based on the work described in [10], where a brush-type model was extended to include the friction measurements, here the model is further developed to predict longitudinal and lateral behaviour of the full tyre over a wider range of normal loads. The novel model results are compared to and validated against flat-track measurements obtained with the same tyre and the same surface (sandpaper) used for the friction measurements.
The paper is laid out as follows. Section 2 introduces the tyre model formulation, together with the considerations of the contact patch shape and the real friction characteristics. Section 3 describes the rubber friction experiments and results, including the creation of the friction master curve integrated into the tyre model. In Section 4, the experimental tyre data used for validation are described. Simulation results obtained with the developed model are presented and compared to the experimental tyre data and corresponding MF6.1 results in Section 5. Conclusions are drawn in Section 6.

Overview of brush model
Owing to its simplicity and physical-parameter approach, the brush-type model was used for the integration of the friction measurements (see Section 4). The basic brush model is an analytical tyre model that describes the tyre as a single line of elastic bristles (or tread elements) protruding from a disk, which is assumed to deform only in the direction normal to the road plane in response to the wheel load to accommodate the contact patch. The bristles -representing the tread, belt and carcass -can move forwards, backwards and sideways across the surface, and are effectively modelled as linear springs [3].
When the tyre is freely rolling, bristles enter the contact patch at the leading edge vertically and remain vertical until they leave at the trailing edge, experiencing no shear forces. Under longitudinal and/or lateral slip, a slip velocity occurs and the base point of the bristles (attached to the disk) will move at a different speed than the bristle tip that contacts the surface. As a result, the bristles are progressively displaced and build up shear forces as they move through the contact patch. The maximum bristle deflection is limited by the available friction between the tyre and the surface, typically determined by the 'static' friction coefficient. Once the maximum deflection value is reached, the bristles start sliding through the rest of the contact patch (see Section 2.3). There are therefore two distinct regions in the contact patch: the adhesion region, and the sliding region. In the adhesion region, the relative position of the bristle tips to their base points is a function of slip, and can be directly related to the adhesive tyre force component. In the sliding region, the generated force depends on the sliding friction coefficient in combination with the normal contact pressure. The total tyre force is a summation of the forces arising in these two regions of the contact patch, see Figure 1. At low slips -when the bristles within the contact patch are mainly adhering to the ground -the major force contributor is the adhesive tyre force, which is primarily determined by the stiffness of the tyre. At medium to high slips -when most of the bristles within the contact patch are sliding -the tyre force is mostly affected by the friction characteristics of the tread rubber.  The maximum available stress, given by the product of the parabolic pressure distribution and the corresponding friction coefficient, is indicated for the µ stick,x,y value (only valid within the adhesion region), and the µ slide (T, v sl ) value (only valid within the sliding region).
The basis for the tyre model developed here is the brush model by [3] that includes two separate friction coefficients, one for the adhesion region (µ stick ) and one for the sliding region (µ slip ). The model will be referred to as the 'two-μ' model. Qualitatively, by introducing two friction coefficients the typical peak in the force-slip curves can be replicated. However, the two-μ model fails to simulate the drop-off in force after the force peak with increasing slip [2,3], as the constant sliding friction coefficient cannot capture the complex nature of rubber friction. To overcome this shortcoming, this paper proposes to replace the discrete µ slip value with real friction characteristics that are dependent on tread temperature, T, and sliding speed, v sl , denoted as µ slide (T, v sl ). The resulting model will be called the 'variable-μ' model and is shown diagrammatically for pure longitudinal and pure lateral slips in Figure 2. Note that µ stick is treated as a constant, and the tread temperature will be provided as a model input from measurement data rather than computed, as explained in Section 2.4. As an alternative to experimental temperature data, the variable-μ model could be combined with a thermal model (e.g. [11]) to align the inputs to the ones used by the MF. Figure 3 shows the main steps for the calculation of the longitudinal and lateral forces used in the variable-μ model. Starting from the inputs (slip ratio, slip angle, normal load, wheel velocity and tread temperature), the model calculates the sliding velocity of the bristle, v sl , in the sliding region (Section 2.3), which together with the tread temperature, T, provides the necessary information to determine µ slide (T, v sl ) from the friction measurement data (Section 2.4). The value of µ slide (T, v sl ) is then used in the calculation of the longitudinal and lateral forces (Section 2.5).

Contact patch considerations
The shape of the contact patch changes with a wide range of variables, including normal load, camber angle, inflation pressure and, crucially for this discussion, slip. Generally speaking, in the longitudinal case, the contact patch shape remains fairly constant (approximately rectangular) with increasing slip ratio [12][13][14], see Figure 4(a). During cornering, the contact patch evolves to a trapezoidal shape with increasing slip angle, see Figure 4(b) [15]. To capture this behaviour, distinctions are made between the longitudinal and lateral slip conditions. For pure braking/traction manoeuvres, the contact patch is assumed consistent across the tyre width (in terms of dimensions and contact pressure distribution, see Section 2.2.1) and, thus, can be modelled by a single row of bristles that changes in length with normal load. For cornering, the variable-μ model considers three distinct bristle rows (or ribs) -A, B and C -of different lengths that vary with slip angle and normal load to account for the trapezoidal contact patch shape, see Section 2.2.2.

Contact pressure distribution
The contact pressure distribution used for the variable-μ model was derived from stationary contact patch measurements obtained with the tyre investigated here. Measurements were taken on a static stiffness test rig together with a high-resolution pressure sensing mat (XSensor IX500 with 1.6 mm spatial resolution). Figure 5 shows an example of the pressure distribution of the upright tyre at a normal load of 12,150 N. At this load, the contact patch is almost square, and exhibits a reasonably even pressure distribution across its width. Pressure peaks can be observed at the left and right sides (approx. at y = ±100 mm), which stem from the share of the normal load carried by the tyre sidewalls. For the sake  of model simplicity, the localised contact pressure peaks will be ignored and a consistent pressure distribution in the lateral direction will be assumed.
The contact pressure distribution along the longitudinal axis of the contact patch at y = 33 mm is shown in Figure 6. When accounting for the decrease in pressure across the tread grooves (at approx. x = ±15, ±45 and ±75 mm), the average pressure across the footprint is reasonably well described by a parabolic pressure distribution. Only towards the leading and trailing edges can a significant deviation between the parabolic assumption and the static measurements be observed. Nonetheless, to keep the model formulation simple and aligned to common brush models found in the literature (e.g. [2][3][4]), the parabolic pressure distribution was deemed sufficiently accurate for this work.

Cornering condition
As mentioned in Section 2.2, the contact patch shape of a side slipping tyre is similar to a trapezoid, which becomes more evident with increasing slip angle. Relevant literature also demonstrates that the contact pressure distribution across the width can no longer be assumed constant. For example, finite element simulations show a contact pressure gradient across the contact patch width [15]. In addition, at very high slip angles, [15] discusses the presence of localised 'lifting' of areas of the contact patch as a result of buckling of the tyre belts, indicating considerable changes in the local contact pressure.
As contact patch measurements of a cornering tyre were not available, indirect data were used to ascertain the contact patch characteristics of the investigated tyre. That is, the tyre temperature recordings across the tread width obtained during cornering tests on a flat-track machine (see Section 4) were analysed.
During the flat-track test, three infrared (IR) sensors are positioned across the tyre width (indicated by A, B and C) to measure the tread surface temperature. Such a temperature evolution is illustrated in Figure 7 for the outward sweep part of the test, i.e. only the part of the test starting from 0°slip angle and sweeping to the maximum/minimum slip angle (approximately ±12°). The test begins by sweeping through negative slip angles. As the tyre is steered, the temperature of the central part of the contact patch (sensor B) and the region towards the 'outside' of the 'corner' (sensor A) increases significantly, while the temperature of the area towards the 'inside' of the corner (sensor C) gradually reduces. Conversely, similar behaviour is seen for the positive slip angle sweep, which was performed directly after the negative slip angle sweep, while the tyre is still 'hot'. With increasing slip, the corner inside tread area (sensor A) steadily decreases in temperature, while the other parts of the contact patch first cool until about 3°slip angle and, then, significantly heat up.
For all tests, the tread centre always shows the highest temperature and reaches very similar temperatures (approx. 80°C to 85°C) at the minimum/maximum slip angle. This behaviour implies that the centre region of the contact patch is influenced by a left/right cornering manoeuvre in a consistent manner and, in turn, its characteristics (dimensions and contact pressure distribution) can be treated independently of slip angle. In contrast, the significant temperature variation between the inner and outer tread regions with positive and negative slip angles indicates considerable changes in the locally generated shear forces and, thus, local normal load and, in turn, the local contact length.
To account for the change of the contact patch characteristics, the variable-μ model divides the contact area into three regions of equal width, each simulated with a row of independent bristles (or 'ribs'), as mentioned in Section 2.2 and shown Figure 4. Effectively, the tyre can be described as three narrow tyres attached to one another. This approach is referred to as the multi-rib contact patch model. Similar approaches have been proposed in the literature (e.g. [2,16,17]). With the multi-rib model, the contact length of the inner and outer rows changes with normal load and slip angle, whereas the centre rib only varies with normal load. For each rib, a parabolic pressure distribution as described in Section 2.2.1 will be assumed.
Assuming a direct relationship between contact area and normal load [17], the share of F z carried by the inner and outer bristle rows is modelled as a function of slip angle. At zero slip, the three ribs share the normal load equally, i.e. the load on each is F z /3. The minimum load on the outer ribs is set to 100 N, to avoid local lift off. In addition, the total normal load across ribs A, B and C is kept constant, i.e.
Thus, the normal load on each rib is given by: where i indicates the individual ribs, which also correspond to the temperature sensors illustrated in Figure 7. The normal load on the outer ribs are assumed to vary linearly with slip angle, α, according to Equations (2) and (3): where j = 1.6 N/rad, which was found empirically through a best-fit approach with the experimental tyre data set (see Section 4).
Corresponding to the normal load variation, the contact lengths of the two outer ribs are assumed to also change with slip angle, while the centre rib is assumed independent of α, i.e.: where L (F z ) is the contact length determined from the MF model used here for comparison purposes, which is given by: where r 0 is the free unloaded tyre radius, c z the vertical stiffness for the nominal tyre load and inflation pressure, and q a1 and q a2 are MF parameters determined during model parameterisation [18]. For the tyre used in this study, c z = 314,931 N/m, r o = 0.3943 m, q a1 = 0.6015, q a2 = 0.8197. It is acknowledged that the above assumptions for F z,i (α) and L i (F z , α) contain considerable simplifications. This model approach is deliberate to keep the formulation simple and to serve as a first approximation allowing the demonstration of the benefits of the integration of friction characteristics. Furthermore, given the complexity of the contact patch geometry under lateral slip conditions, only pure longitudinal and pure lateral slip cases are examined in this paper. For combined slip cases, the contact patch geometry is more complex still, and requires further analysis.

Bristle sliding velocity
As discussed in Section 2.1, when a tread element enters the contact patch, it adheres to the ground and only starts to slide when the local friction force capacity is reached. The onset of sliding is a highly dynamic event that is influenced by several parameters such as the friction coefficient, the contact pressure, the velocity and the strain energy stored in the tread element [19]. However, to keep the formulation simple, similar to the TreadSim model [2], the sliding velocity is assumed to be equal to the velocity of the base point of the tread element, i.e. the slip velocity. For pure longitudinal motion, in the absence of camber, ply steer or conicity effects, the sliding velocity is given by: where v x is the translational velocity of the wheel hub and κ is the practical longitudinal slip [2]. In the lateral case, the sliding velocity is given by: Once the sliding velocity of the tread element is known, the µ slide (T, v sl ) value can be determined for a given tread temperature, as explained in Section 2.4.

Friction master curve
The real tread rubber friction characteristics are considered by incorporating the friction master curve in the variable-μ model. Details on how the master curve is constructed are discussed in Section 3, while its implementation is presented here. The friction master curve describes how the friction coefficient changes with sliding speed at a constant temperature, i.e. the reference temperature, T s [20]. In reality, maintaining a constant temperature at sliding speeds exceeding approximately 10 mm/s is not possible, due to local heating effects [7,20]. Thus, to allow simulation of different tyre temperatures, the friction master curve needs to be integrated with a temperaturedependent shift factor that allows replicating the friction characteristics at an arbitrary temperature, T'.
The shift method integrated in the variable-μ model is an extension of the WLF transform [20,21,25] used to create the friction master curve (see Section 3), and is given by: where a T is the shift factor and T g is the glass transition temperature of the rubber. Figure 8 illustrates the shift procedure, and shows how temperature can significantly affect the friction coefficient, µ slide (T, v sl ). For example, for a given sliding velocity, v sl,1 , shifting the master curve to a higher temperature causes a significant reduction in the corresponding friction value. Indeed, depending on the sliding velocity, the friction coefficient can decrease, so that µ slide (T', v sl,1 ) < µ slide (T s , v sl,1 ), or increase, so that µ slide (T', v sl,2 ) > µ slide (T s , v sl,2 ).
Thus, temperature is an important input in the variable-μ model and, because of its importance, the tread temperature will be provided as an input to the model rather than computed to avoid introducing associated modelling inaccuracies. For the simulation of the longitudinal slip characteristics that will be compared to experiments and MF results in Section 5.1, the tread temperature recording from the centre sensor (sensor B) during the experiments will be used. For the lateral slip characteristics (see Section 5.2), the temperature readings from the three temperature sensors (A, B and C) will be used for the force computation within the corresponding tread rib.
It should be noted that the commonly observed load-dependency of tyres is captured by the shifting of the master curve. Changes in the contact pressure lead to changes in frictional heating in the rubber, which, in turn, influences the tyre-surface friction coefficient and thus the tyre shear forces [26].

Force calculation
Using the formulation of the two-μ brush model by [3], equations for the longitudinal and lateral forces can be derived for steady-state conditions. First, the shear stress, τ , built up in the bristle can be described by: where G is the shear modulus of the rubber and γ is the shear strain, which is given by the ratio of the bristle displacement and the (undeformed) bristle height. It is assumed that G x = G y = G. Using the bristle position, ξ (measured from the leading edge, ξ = L/2 − x), the shear strain as a function of position in the contact patch reads: where v is the slip speed, R is the rolling radius, t(ξ ) = ξ/|Rω| is the time taken to reach position ξ in the contact patch with respect to the leading edge, and H is the (undeformed) tread depth. Combining Equations (9) and (10), one can write the shear stress as a function of position in the contact patch as: where σ x,y is the theoretical slip [2]. In the lateral case: which, for the case of pure cornering is equal to the practical slip: In the longitudinal case: It should be noted that σ x is different to the practical slip in Equation (6). The two are related by: Thus, for the purposes of integration of flat-track data, which uses a practical slip approach, the theoretical slip is used for the force calculation, but results are plotted in terms of practical slip.
With this definition of the shear stress, one can compute the force, F x,y , by integrating the shear stress over the contact patch area, A (with A = WL, where W is the contact patch width): If the friction limit is not reached within the contact patch: where C x,y is the longitudinal or lateral slip stiffness, which becomes: If the friction limit is reached in the contact patch, the so-called 'breakaway point', ξ c (i.e. the transition from the adhesion to sliding region of the contact patch) is found by equating the shear stress at co-ordinate ξ c to the maximum available frictional stress: where µ stick,x,y is the friction coefficient in the adhesion region (cf. Section 2.1), and p(ξ ) is the local contact pressure. Here, the contact pressure is assumed to follow a parabolic distribution along the contact length: Hence: The total tyre force is then given by two shear force contributions: F x,y,stick , which calculates the force contribution from the leading edge up to ξ c , and F x,y,slide , which calculates the force contribution from ξ c to L, where the stress equals µ slide (T, v sl ) p(ξ ): (23) Solving the integral (23), and considering that the sliding region extends over the entire contact patch when ξ c = 0 (i.e. |σ x | > 3μ stick,x,y F z C x,y ), the longitudinal force becomes: Similarly, the lateral force is given by Equation (25), noting that C y and F z are functions of slip angle, α, for the multi-rib model (see Section 2.2.2): As rubber has a complex shear modulus [22], instead of calculating C x and C y from (18) and (19), the relevant stiffness values were directly derived from the experimental flat-track data, which will be denoted as C x,y,exp . C x,exp was found from the force gradient over a ±1% longitudinal slip range and C y,exp over a ±1°slip angle, respectively.
Equation (19) also shows that C y depends on the length and width of the contact patch. Hence, for the multi-rib model (cf. Section 2.2.2), the lateral slip stiffness is computed for each rib. Since each rib has equal width, but a varying contact length L i (F z , α), the lateral slip stiffness per rib reads: where L i (F z , α) is given by Equation (4) and L(F z ) is obtained from Equation (5).

Rubber friction testing
A state-of-the-art, purpose-built test rig (Figure 9) was used to measure the rubber friction characteristics. The rig consists of a linear actuator that moves a surface at a user-defined velocity within a climate chamber. A tread rubber sample, fixed in a sample holder, is pressed against the test surface by weights while the frictional, normal and lateral forces are measured. Similar to the fundamental work by Grosch [20], measurements were carried out at sliding velocities ≤ 10 mm/s to avoid frictional heating [7,20]. By repeating friction experiments at different temperatures, the measurements allow for the construction of a friction master curve as explained in Section 3.2. Measurements were carried out on the same sandpaper surface as installed on the flat-track test rig that was used to obtain the real tyre force slip characteristics (see Section 4) for the assessment of the accuracy of the variable-μ model. The rubber sample was cut from a new tyre ( Figure 10). To ensure consistent and repeatable results, a robust sample preparation and run-in procedure was carried out before friction data were recorded. This consisted of wearing-in the sample by, firstly, handsanding to remove the outer layer and, secondly, running it at a constant velocity in the rig for several hours until a stable friction coefficient was reached. A more detailed account of the sample preparation is given in [10]. After a sample was run-in, test runs began. A test run is defined as the sample traversing a 70 mm section of the sandpaper surface at 14 different velocities, ranging from 0.03 to 10 mm/s, at one temperature. For each velocity in a test run, the sample traversed the 70 mm section back and forth twice, and the friction coefficient measurement was taken as the average value across this distance.
Test runs were carried out at 16 different temperatures ranging from −40°C to +50°C. During a test run, the sliding velocity was changed from the lowest to the highest one to minimise the effects of sample wear. For example, at cold temperatures in particular, wear was greater for higher sliding velocities, evident by the observable debris on the surface after the test, see Figure 10. The surface and the sample were cleaned after each test run. In addition, testing was peppered with 'calibration runs', where the sample was slid at the run-in speed to monitor consistency and repeatability of the measurements. Tests were also repeated over a period of months to check and ensure consistency and repeatability.

Friction measurement results
Following the test procedure outlined in Section 3.1, a series of friction curves over a narrow velocity range (between 0.03 and 10 mm/s) and at different constant temperatures was obtained, see Figure 11. As the very low sliding velocities, v, of the friction tests are not representative of real tyre contact patch sliding speeds, the individual friction curves were merged to form the friction master curve. Similar to the fundamental work by Grosch [20], the Williams-Landel-Ferry (WLF) transform was used.  The WLF provides a temperature-dependent transformation factor, a T , that can be applied to each friction curve by plotting the measured µ as a function of (a T ·v), rather than v. As a result, the individual curves are shifted over a wide range of sliding velocities forming a single 'master curve', but at a specific reference temperature, T s . The WLF is given by: where T is the temperature of the curve which is being shifted. The reference temperature is set at 50°C above the glass transition temperature, T g , of the rubber compound [20]. T g was determined through DMA testing using three tread compound samples. Similar to the literature (e.g. [22][23][24]), here, T g was taken as the temperature corresponding to the peak in the experimentally determined tan δ curve. By averaging the results from all samples, T g was found to be equal to −30°C. Thus, T s = 20°C. Figure 12 shows the individual friction measurements shifted to the reference temperature yielding a very wide sliding velocity range, plotted as log(a T ·v). The master curve was approximated from the measurements by fitting a spline, which is shown by the solid red line. The spline was integrated into the variable-μ brush model as explained in Section 2.4.

Experimental tyre data
To assess the accuracy of the developed variable-μ model, simulation results are compared to experimental tyre data obtained on an indoor flat-track test rig at Calspan (USA), and the corresponding Magic Formula model. In this paper, only pure slip condition tests with an upright tyre at one inflation pressure are presented. Due to the nature of the flat-track procedure, a different number of normal loads were tested for longitudinal and lateral slip tests (five and three, respectively).
For both longitudinal and lateral tests, the input to the flat-track machine for the F z value was kept constant, while the slip ratio or slip angle was changed. Tests were conducted at a constant velocity v x = 16.7 m/s, on a 120-grit sandpaper belt -the same sandpaper that was used for friction testing described in Section 3.
Flat-track data were post-processed to ensure reliable comparison with the variable-μ brush model. Post-processing consisted of cropping data so that only the 'outbound' sweeps were retained. That is, sweeps from 0 slip to maximum/minimum slip were retained; data of the 'return' sweep from the slip extrema back to 0 were discarded. The primary reason for this was to ensure the validity of the WLF transform, which is only applicable for temperatures of up to 100°C above T g [20], i.e. approx. 70°C. Since the tyre undergoes significant energy dissipation at high slips, the tyre was often heated to temperatures outside of this range at the maximum/minimum slip, and did not cool down until it had passed through 0 slip, before undergoing the remainder of the test. An example of the cropping process of the dataset is shown in Figure 13. Variable-μ model and MF results (see Section 5) were compared to the flat-track data, using the root mean squared error (RMSE): where k is a data point, N is the maximum number of data points, m is the model output and f is the flat-track measurement.

Longitudinal slip characteristics
The normal loads for which flat-track data were extracted are shown in Table 1, alongside the values of C x,exp and µ stick,x used in the variable-μ model. C x,exp was calculated from the force-slip gradient of the flat-track data (see Section 2.5), while µ stick,x was empirically obtained by fitting the model results to the experimental tyre data.
To visualise the summation from the adhesive and sliding regions of the contact patch, the force contribution from each region is shown separately (and the two summed together) in Figure 14, together with the flat-track data, two-μ model (with μ stick,x of Table 1 and µ slide = 1.25), and the corresponding MF6.1 model results. The variable-μ model predicts tyre behaviour well, capturing the peak force and drop-off accurately, unlike the two-μ model as discussed in Section 2.1. For negative slips, it can be argued that the variable-μ model predicts tyre behaviour better than the MF. For example, utilising Equation (28), the RMSE for κ < 0 was 225 N for the variable-μ model, compared with 270 N for the MF. After the peak force for κ < 0, the RMSE was 237 N for the variable-μ model compared with 275 N for the MF. Figure 14 also illustrates how the relative contributions from the two contact patch regions affect the overall force characteristics: the size of the adhesion region affect the slip stiffness, while the frictional characteristics of tyre-road interaction affect the peak and drop-off forces. Interestingly, the slight asymmetry around the peak force seen in the flat-track data with positive and negative slip ratios -primarily a temperature effectcan be captured through the integration of the measured friction characteristics.  Table 1 obtained with the MF model (blue line) and the variable-μ model, which are coloured by the tread temperature (sensor B, cf. Figure 7). Experimental flat-track data are given by the black dotted lines.
The longitudinal slip characteristics at five different normal loads are collated in Figure  15. In general, compared to the experimental data, the variable-μ model describes the longitudinal force against slip ratio behaviour well for a wide range of normal loads and even at elevated temperatures beyond the commonly assumed validity of the WLF transform. Also, for many slip conditions, the variable-μ model matches the tyre data better than the MF model. For example, the RMSE value for κ > 0.1 (corresponding to the peak force) at 7 kN was 116 N for the variable-μ model, versus 214 N for the MF. However, at the highest investigated normal load (13, 757 N), the variable-μ model displays a significant overshoot around the peak force. This model discrepancy is due to the extreme test condition with a normal load that exceeds the rated load of the tested tyre (1060 kg). As a consequence, during the test, large deformations occurred within the tyre structure, yielding geometric nonlinearities that cannot be captured by the simple contact patch formulation of the variable-μ model.

Lateral slip characteristics
As discussed in Section 2.2.2, the multi-rib contact patch model was used for the lateral force calculation. The corresponding model parameters in terms of C y,exp and µ stick,y are shown in Table 2. To illustrate the multi-rib approach, Figure 16 shows the force contribution of each rib in the calculation of the total force together with experimental results. The discontinuity of the experimental data for small slip angles is a consequence of the post-processing outlined in Section 4. Similar to the longitudinal slip characteristics, the variable-μ model matches the lateral force experiments well, especially for the negative slip sweep. For the positive slip sweep, which was carried out directly after the positive slip angle sweep, small differences can be observed for slip angles approx. > 6°. The difference  can be primarily ascribed to a combination of elevated temperatures during the test and tyre structure nonlinearities, i.e. ply-steer and conicity [2]. Figure 17 compares the simulation results at three different normal loads with the experimental data and the MF model. Generally, the variable-μ model shows a satisfactory correlation at the different loads. It can match the cornering stiffness well for all loads and better than the MF model, but shows differences in the simulated lateral force for slip angles greater than about 5°with increasing normal load. Again, the model mismatch is more evident for the positive slip angles than for negative slip angles.
Similar to the longitudinal slip characteristics, the variable-μ model shows a significant overshoot compared to the experiments and the MF results at the highest normal load (9 kN), resulting in an RMSE value of 947 N for the variable-μ model vs 225 N for the MF for α ≤ −6 • . This behaviour can be mostly ascribed to nonlinear tyre deformation behaviour which is more complex compared to the longitudinal manoeuvre because of the off-axis resultant force vector. That is, during cornering, the contact patch will be deformed laterally (twisted) and longitudinally (stretched). Also, the model discrepancy demonstrates that the tyre force nonlinearities at high slip are not only due to friction influences, but also due to carcass/tyre structure effects, which are not accounted for in the contact patch model of the variable-μ model. These effects -in conjunction with the limited validity of the WLF transform (cf. Section 4) -also account for the uptick in predicted lateral forces at the highest slip angles at the 6 and 9 kN loads. In sum, the close match of the results demonstrates the benefit of the inclusion of the real friction characteristics and the observations illustrate the benefits from employing a physical modelling approach.

Conclusions
In this paper, friction measurements with a tread rubber block slid against a sandpaper surface were performed under controlled conditions and integrated with a simple, low-parameter brush model, producing the so-called 'variable-μ model'. Comparison of the model results with experimental slip characteristics obtained on the same sandpaper surface and the corresponding MF model allow the following conclusions: • Despite the very simple, low-parameter brush model approach used here, the integration of the real friction characteristics allows very good replication of the longitudinal tyre behaviour over a wide range of operating conditions in terms of normal load and slip ratio. For lateral slip characteristics, good agreement between simulations and experiments was observed for low to medium vertical loads. • With very high normal loads, the variable-μ model overpredicted the experiments due to nonlinear effects arising from large tyre deformations that are not captured in the simple contact patch formulation adopted here. • Compared to the MF model, requiring a total of 42 parameters to simulate pure longitudinal and pure lateral slip conditions [18], the variable-μ model contains considerably fewer parameters -12 parameters or 25 if modelling of the temperature according to [11] would be considered. Nonetheless, the variable-μ model can replicate very similar behaviour for the longitudinal force characteristics and reasonable behaviour for lateral slip behaviour. While not seeking to replace the MF, the proposed approach shows the value and importance in capturing accurate friction characteristics for tyre modelling. • Under certain operating conditions the variable-μ model can provide a better match with the experiments than the MF model; for example, this includes small slip angle and large slip ratio conditions.
In addition, the information associated with the physical modelling approach (e.g. breakdown of the contact patch contributions) was shown to provide valuable insights into tyre behaviour, which is not available using the empirical MF.

Disclosure statement
No potential conflict of interest was reported by the author(s).