Question

Let's revisit our linear inverted pendulum (LIP) model for analyzing the centerof-mass (CoM) dynamics of bipedal systems. Recall the dynamics of the LIP: x¨=ω2(x−p), where x is the CoM position, p is the center of pressure (CoR) position, and ω is the natural frequency of the pendulum. As previously shown, the CoM dynamics have a stable component of motion, and a divergent component of motion (see the characteristic equation from HW 3 Problem 4 (a)). Let's say we were again developing a balancing controller for a bipedal robot and we want the controller to be robust to disturbances (pushes to the robot's CoM). Let's investigate the unstable system pole cancellation method of control. (a) (10 pts) Derive the transfer function for the LIP dynamics. Hint: Recall the CoP is the input to the LTI system. (b) (10 pts) Select P(s) such that it cancels the unstable component of the CoM dynamics. If the robot is pushed, does it stabilize itself? Now assume there is a measurement error in our CoM height H calculation, maybe 0.001 m (but our controller is unaware). Can the robot still stabilize itself? Is our control method robust to parameter estimation error? Once again use your own CoM height for the analysis. Hint: Use the MATLAB function impulse() to simulate the system in the sdomain. We are interested in impulse response analysis.

Answer 1

The transfer function for the linear inverted pendulum (LIP) dynamics can be derived using the equation of motion. The transfer function is given by G(s) = 1/(s^2 - ω^2), where s is the Laplace variable and ω is the natural frequency of the pendulum.

Step-by-step explanation:

To derive the transfer function for the linear inverted pendulum (LIP) dynamics, we start with the equation of motion for the CoM position, which is given by x¨ = ω^2 (x - p), where x represents the CoM position, p represents the center of pressure (CoR) position, and ω is the natural frequency of the pendulum.

We can rearrange the equation in terms of the Laplace variable, s, and define the transfer function, G(s), as the ratio of the Laplace transform of the CoM position, X(s), to the Laplace transform of the CoR position, P(s).

By taking the Laplace transform of the equation of motion and rearranging, we obtain the transfer function: G(s) = X(s)/P(s) = 1/(s^2 - ω^2). This transfer function represents the LIP dynamics.