means of a second hydraulic cylinder 50 a bucket 56 is pivotally connected at the end of stick 48 at a joint 58 the angle u03b8 3 fig2 between stick 48 and bucket 56 may be adjusted by means of a third hydraulic cylinder 60 backhoe 20 may be considered to have four degrees of freedom these are angles u03b8 1 and u03b8 2 which together adjust the position of the end of stick 48 in the vertical plane of arm 41 angle u03b8 3 which adjusts the orientation of bucket 56 i e the angle u03b3 of fig2 and the swing angle u03c6 which may be measured relative to an arbitrary reference line such as longitudinal axis 26 backhoe 20 includes a power source 21 fig7 typically power source 21 comprises a diesel engine driving one or more hydraulic pumps the hydraulic pumps produce a supply of pressurized hydraulic fluid valves 81 allow the supply of pressurized hydraulic fluid to be selectively connected to actuators such as actuator 34 and hydraulic cylinders 44 50 and 60 in some cases power source 21 could be a source of electrical power in such cases actuator 34 and hydraulic cylinders 44 50 and 60 would be replaced with electrical actuators and valves 81 would be replaced by electrical switches as shown in fig2 the position of a point 65 at the tip of bucket 56 may be specified in a cylindrical frame of reference f arm centred on axis 33 in this frame of reference the position vector p of point 65 may be specified in terms of the coordinates r z and u03c6 as shown in fig2 in the alternative the position of point 65 relative to.