Computed torque control explained

Computed torque control is a control scheme used in motion control in robotics. It combines feedback linearization via a PID controller of the error with a dynamical model of the controlled robot.^{[1] [2]}

Let the dynamics of the controlled robot be described by

M \left( \vec\theta \right) \ddot\vec\theta + C \left( \vec\theta, • \vec\theta \right) • \vec\theta + \vec\tau _{g} \left(\vec\theta\right) = \vec\tau where \vec\theta \in R ^{N} is the state vector of joint variables that describe the system, M \left(\vec\theta\right) is the inertia matrix, C \left( \vec\theta, • \vec\theta \right) • \vec\theta is the vector Coriolis and centrifugal torques, \vec\tau _{g} \left(\vec\theta\right) are the torques caused by gravity and \vec\tau is the vector of joint torque inputs.

Assume that we have an approximate model of the system made up of

\tilde M \left( \vec\theta \right), \tilde C \left( \vec\theta, • \vec\theta \right), \tilde\vec\tau _{g} \left(\vec\theta\right) . This model does not need to be perfect, but it should justify the approximations M \left( \vec\theta \right) ^{-1} \tilde M \left( \vec\theta \right) ≈ 1 and M ^{-1} \left( C \left( \vec\theta, • \vec\theta \right) • \vec\theta + \vec\tau _{g} \left(\vec\theta\right) \right) ≈ M ^{-1} \left(\tilde C \left( \vec\theta, • \vec\theta \right) • \vec\theta + \tilde\vec\tau _{g} \left(\vec\theta\right) \right) .

Given a desired trajectory

\vec\theta _{d(t)} the error relative to the current state \vec\theta(t) is then \vec\theta _{e(t)} = \vec\theta _{d(t)} - \vec\theta(t) .

We can then set the input of the system to be

\vec\tau(t) = \tilde M \left( \vec\theta \right) \left( \ddot\vec\theta _{d(t)} + K _{p} \vec\theta _{e(t)} + K _{i} \int ^{t} _{0} \ddot\vec\theta _{e(t')} dt' + K _{d} • \vec\theta _{e(t) \right)} + \tilde C \left( \vec\theta, • \vec\theta \right) + \tilde\vec\tau _{g} \left(\vec\theta\right)

With this input the dynamics of the entire systems becomes

\begin{align} M \left( \vec\theta \right) \ddot\vec\theta + C \left( \vec\theta, • \vec\theta \right) • \vec\theta + \vec\tau _{g} \left(\vec\theta\right) = & \tilde M \left( \vec\theta \right) \left( \ddot\vec\theta _{d(t)} + K _{p} \vec\theta _{e(t)} + K _{i} \int ^{t} _{0} \ddot\vec\theta _{e(t')} dt' + K _{d} • \vec\theta _{e(t) \right)} + \tilde C \left( \vec\theta, • \vec\theta \right) + \tilde\vec\tau _{g} \left(\vec\theta\right) \\ \ddot\vec\theta + M \left( \vec\theta \right) ^{-1} \left( C \left( \vec\theta, • \vec\theta \right) • \vec\theta + \vec\tau _{g} \left(\vec\theta\right) \right) = & \underbrace{ M \left( \vec\theta \right) ^{-1} \tilde M \left( \vec\theta \right) } _{≈} \left( \ddot\vec\theta _{d(t)} + K _{p} \vec\theta _{e(t)} + K _{i} \int ^{t} _{0} \ddot\vec\theta _{e(t')} dt' + K _{d} • \vec\theta _{e(t) \right)} + M \left( \vec\theta \right) ^{-1} \left(\tilde C \left( \vec\theta, • \vec\theta \right) + \tilde\vec\tau _{g} \left(\vec\theta\right) \right) \\ \ddot\vec\theta = & \ddot\vec\theta _{d(t)} + K _{p} \vec\theta _{e(t)} + K _{i} \int ^{t} _{0} \ddot\vec\theta _{e(t')} dt' + K _{d} • \vec\theta _{e(t) \\ 0} = & \ddot\vec\theta _{e +} K _{p} \vec\theta _{e(t)} + K _{i} \int ^{t} _{0} \ddot\vec\theta _{e(t')} dt' + K _{d} • \vec\theta _{e(t) \end{align}}

and the normal methods for PID controller tuning can be applied. In this way the complicated nonlinear control problem has been reduced to a relatively simple linear control problem.

Notes and References

1. Book: Lynch . Kevin M. . Modern robotics: mechanics, planning, and control . Park . Frank C. . 2017 . Cambridge university press . 978-1-107-15630-2 . Cambridge.
2. Book: Springer handbook of robotics . 2016 . Springer . 978-3-319-32550-7 . Siciliano . Bruno . 2nd . Springer handbooks . Berlin Heidelberg . 174–175 . Khatib . Oussama.