computes states the lead to violations of the lyapunov conditions @return z_adv_pos tensor [num_samples, x/z_dim] where the lyapunov

The apex happens when ydot crosses zero from above.

Given the apex state [x;z;ẋ], and the leg angle, find the linear approximation of the apex-to-apex return map. Notice that here in t

@return: r: numpy array of size (nx, nx). immediate cost from state x to x'. u_optimal: numpy array of size (nx, nx, u_dim). u[i, j] is t

@return: r: numpy array of size (nx, nx). immediate cost from state x to x'. u_optimal: numpy array of size (nx, nx, u_dim). u[i, j] is t

Compute the cost-to-go for a continuous time piecewise affine system. The cost-to-go is defined as V(x) = ∫ᵀ₀ cost(x(t))dt We will fi

Compute the cost-to-go ∑ᵢ c(x[i]) starting from x_start for the discrete-time system. If the trajectory of x reaches x_goal, or after sim

A simple controller mimicking optical flow controller to avoid obstacles. Assume field of view [0,pi/2) Make control decisions only based

Driving to the goal position while avoiding obstacles. Assume field of view [0,pi/2). Make control decisions only based on views in the f

Compute z and Wz defined in output_gradient_times_vector() function. Notice that when the relu network input is 0, we use only the right

Construct a trajectory optimization problem to find the minimal time to reach the goal position. The user should adjusts initial_val_constrai

Creates plant that includes only the robot, used for controllers. :param gravity: :return:

The double integrator p̈ = u can be written as a linear system ẋ = Ax + Bu where x = [p, ṗ]. A = [0 1; 0 0], B = [0; 1] @return (A,

Compute the LQR controller for double integrator. The LQR controller is u = -R⁻¹BᵀPx where P is the solution to the Riccati equation

For a feedforward network with (leaky) relu activation units, extract the weights and bias into one tensor.

The dynamics is ẋ=f(x)+G(x)u

Generate the dataset to train a controller and value function. Simulate the pendulum dynamics (and cost-to-go) using an energy shaping +

Generate many pairs of state/action to next state. Notice that since the car dynamics is shift invariant, our input doesn't include the c

Generate the pairs (x[n], u[n]) -> (x[n+1])

Generate the pairs (x[n], u[n]) -> (x[n+1])

return a list of lists of parameters that correpond to the lyapunov only

return a list of lists of parameters that are trainable

Test lyapunov_derivative_as_milp function, but with a lower and upper bounds on V(x[n])

We will sample states x̅ⁱ, i=1,...N, and we would like the Lyapunov function to decrease on these sampled states x̅ⁱ. We denote l(x)

We will sample states x̅ⁱ, i=1,...N, and we would like the Lyapunov function to decrease on these sampled states x̅ⁱ. We denote l(x)

We can rewrite the hybrid dynamics as mixed integer linear constraints. We denote αᵢ = 1 if the system is in mode i. x[n+1] =

We can rewrite the hybrid dynamics as mixed integer linear constraints. We denote γᵢ = 1 if the system is in mode i. ẋ = ∑ᵢ

@return mixed-integer linear constraints MixedIntegerConstraintsReturn Ain_x, Ain_s, Ain_gamma, rhs_in, Aeq_x

@return mixed-integer linear constraints MixedIntegerConstraintsReturn Ain_x, Ain_s, Ain_gamma, rhs_in, Aeq_x

@return mixed-integer linear constraints MixedIntegerConstraintsReturn Ain_x, Ain_s, Ain_gamma, rhs_in, Aeq_x

Returns the mode of state x. Namely P[mode] * x <= q[mode]. returns None if x is not in any mode. Notice that we choose the f

Returns the index of the node with the nearest distance to x, measured as the weighted L2 distance with weights Q.

Return the indices of the nodes within distance radius to x.

Compute the next pose (x_next, y_next, yaw_next) given the current state (x, y, yaw, vel) and control (yaw_rate, accel) after dt.

Computes the next pose of the quadrotor after dt.

The ReLU network output is a piecewise linear function of the input x. Hence the gradient of the output w.r.t the input can be expres

generates a rollout with the learned dynamics for N step @param x_init tensor of dim [2*num_channels, width, height] @param N