The diagram shown on the next page represents a planar pantograph-based leg for a walking robot. This model utilizes only one DOF to generate the walking gait at the foot link 'n'. One rotary actuator drives joint 'q', while joints 'O' & 'U' are allowed to rotate about fixed axes.

• You are to obtain the mathematical model equations of the leg's forward kinematics and to verify this model using simulation.

v Suggested Procedure: 

1)    Assign appropriate link frames.  Note that:

a)     This mechanism is a closed link chain.

b)    There are two fixed frames:  { U } & { O }.

c)     Joint angle 'q ' is the input variable (one DOF).

d)    Link lengths a, b, c, d, q and the distance 'm' are known.

2)    Obtain the forward kinematics equation :   [ ^O_nT(q) ] , then find the  expressions for the position vector at the origin of the foot frame ( ^OP_norg  ) and the orientation of this frame ( ^O_nR ).

3)      Write a Matlab routine to calculate the Cartesian trajectory of the foot position in response to one rotation of the input joint 'q'.  The routine should plot the input and output trajectories in Cartesian space.  You are free to use the link dimensions provided on the next page or select your own.  Verify the resulting amplification factor of this mechanism in either case.