9. Appendix A - Kinematic CalculationsHexotica - The Design and Implementation of a Small Walking Robot
9. Appendix A - Kinematic Calculations
In order to calculate the position of the end effector of the foot, three frames of reference can be attached to the leg, as shown in Figure 51. Frame 0, denoted Á 0, is considered to be an inertial frame of reference for this calculation. Note that this frame will, in fact, be attached to the robot and will thus move with the robot. For present purposes, however, it can be considered to be an inertial frame. Frame 1 is attained via a rotation about the Y0 axis through some angle b . Frame 2 results from a rotation of g about Z1 and Frame 3 is attained via a translation of L1 along the X2 axis followed by a rotation about Z2 of -(180-q ).
Figure 51 - Frames of reference used for end effector calculation
The actual position of the foot can be calculated by describing the vector to the end point in the inertial frame of the leg. This is done as follows.
Equation 6 - Derivation of the position of the foot
By taking the partial derivative of the positions with respect to each of the joint angles, the Jacobian matrix can be derived. This matrix can be used to relate Cartesian velocities of the foot to the joint velocities of the leg. The derivation of this relationship is shown in Equation 7. The Jacobian matrix and its inverse are shown in Equation 8 and Equation 9 respectively.
Equation 7 - Derivation of transformation from Cartesian velocities to joint velocities
Equation 8 - Jacobian matrix relating Cartesian positions with angles
Equation 9 - Inverse of the Jacobian used to transform Cartesian velocities to joint velocities
As mentioned in the discussion of the new leg configuration, an approximation of the relationship between the driving length, l , and the controlled angle, q , is needed for control purposes. This approximation is necessary because a lead screw is being used to control an angular position at the remote knee joint. This relationship is shown in Figure 52.
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