PROBABILISTIC MOTION ESTIMATION

This Matlab toolbox implements the algorithms developed here. Given a database of motion capture data, the toolbox estimates multi-joint movement trajectories as well as constant parameters such as limb sizes, axes of joint rotation, marker positions and orientations relative to the underlying limb segments. The computation is fully probabilistic and yields not only the most likely values but also their confidence intervals. The entire toolbox, along with a user's manual and numerical examples, can be downloaded as a single zip archive.

ITERATIVE LQG CONTROL OF NONLINEAR SYSTEMS

The Matlab function ilqg_det.m implements the deterministic case of the algorithm developed here. Given a dynamical system and a cost function, it constructs a locally-optimal feedback control law via iterated LQG approximations. Its use in the context of a center-out reaching task (formulated in joint space) is illustrated in the script test_ilqg_det.m. An implementation for stochastic systems will be available soon.

Among the arguments of ilqg_det are handles to Matlab functions computing the dynamics, cost, and their derivatives with respect to the state and control variables. If the derivative calculation cannot be done analytically one should implement a finite difference method. The format of the dynamics function is illustrated in arm_dyn.m which computes the dynamics of a 2-link torque-controlled arm moving in the horizontal plane. The format of the cost function is illustrated in arm_cost.m which computes a weighted sum of control energy and endpoint error. Both of these functions are needed to run test_ilqg_det.

ESTIMATION AND CONTROL WITH SIGNAL-DEPENDENT NOISE

The Matlab function kalman_lqg.m implements the algorithm developed here. It constructs a modified Kalman filter + LQG controller pair for a linear-quadratic system subject to a combination of additive, state-dependent, control-dependent and internal noise. The function can also simulate a specified number of noisy trajectories. Its application in the context of reaching movements is illustrated in the script test_kalman_lqg.m.

The present algorithm is a generalization of the algorithm we used to construct our optimal feedback control models of motor coordination. The earlier version of the algorithm did not allow state-dependent and internal noise. The latter are useful in modeling active sensors and memory decay respectively.

MINIMUM JERK TRAJECTORIES

The Matlab function min_jerk.m implements the algorithm developed here. Given a sequence of points in 2D or 3D, the velocities and accelerations at the two endpoints, and the movement duration, the function computes the minimum-jerk trajectory (that is, the trajectory minimizing the integral of the squared derivative of acceleration). When the passage times through the intermediate points are not specified the function optimizes over them using the nonlinear simplex method. This is particularly useful when a movement path is given and one wants to compute a reasonable speed profile.

The script test_min_jerk.m illustrates the use of the function in 2D. It waits for the user to enter a sequence of points by clicking over a figure. Pressing Enter signals the end of the sequence. Note that the points are not visible while they are being entered.

BACK-PROPAGATION

The Matlab function backprop.m is an efficient implementation of the backpropagation algorithm for computing gradients in feedforward neural networks. It avoids loops over the dataset and handles arbitrary network topologies. The topology is defined by a square matrix whose entries indicate the connectivity between all pairs of neurons. The transfer functions of the neurons can be set independently (via a vector of flags) to sigmoid, tanh, soft-threshold-linear, or linear.

Backpropagation computes the gradient of the error with respect to the weights, but in itself is not a learning algorithm capable of improving the weights. To implement learning one must couple backpropagation with gradient descent. This is illustrated in the script test_backprop.m. For more efficient optimization we recommend the function minimize.m written by Carl Rasmussen and available on his website.