Knee torque of a lumped mass model

Knee joint in a lumped mass model

Consider the planar legged model depicted to the right. The robot's mass is concentrated at its center of mass \(G\), meaning we neglect the mass of the two links of the leg. The center of mass is located above the center of pressure \(C\) and the robot is in static equilibrium. The leg has a single revolute joint located at the knee \(K\) with joint angle \(q\). Both of its links have the same length \(CK = KG = \ell\).

Question: what is the joint torque \(\tau\) exerted by the knee to keep the leg in static equilibrium?

Note

This is a question page: there is no need to post the answer in the discussion below. However, if you have an original derivation of the solution, you are welcome to post it.

To go further

This simplified model was considered to dimension the knee actuators during the design of the HRP-4 humanoid robot.

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