Question 1(40%) Figure 1 describes a planar robot comprised of a moving cart with telescopic arm, working under gravity.

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Question 1 40 Figure 1 Describes A Planar Robot Comprised Of A Moving Cart With Telescopic Arm Working Under Gravity 1 (399.13 KiB) Viewed 41 times

Question 1(40%) Figure 1 describes a planar robot comprised of a moving cart with telescopic arm, working under gravity. The vector of joint variables is q=(d1​,θ2​,d3​)T. The robot carries a tool at the end effector, with mass mt​ and tensor of inertia Itt​=⎣⎡​I1​00​0I2​0​00I3​​⎦⎤​, expressed in the frame (x^t​,y^​t​,z^t​) attached to the tool. The tool's center-of mass is located at the end effector's center. The mass of the cart is M and the masses of all other links are negligible. 1.1 Solve the robot's forward kinematics - write the homogeneous transformation matrix 0 At​ for given joint variables q=(d1​,θ2​,d3​)T. Use the definitions of joint variables and reference frames as given in Figure 1. 1.2 Write the matrix of inertia H(q) and vector of gravitational terms G(q) which appear in the robot's dynamic equation of motion H(q)q˙​+C(q,q˙​)q˙​+G(q)=τ. Explain all stages of derivation in detail. No need to write the velocity-dependent terms C(q,q˙​). 1.3 In order to prevent the cart from tipping over, it is required that the normal reaction force (in direction y^​0​ ) applied on it by the ground is positive. Write this requirement as an explicit inequality expressed in terms of joint variables, their velocities and accelerations q, q,q¨​.