State-space models and capstone: a motor-driven joint

Transfer functions describe how one input drives one output, and they hide everything that happens inside. Robots, however, have many inputs and outputs, internal variables that matter in their own right, such as motor current and link velocity, and initial conditions that cannot be ignored. The state-space model handles all of this. It writes a system as a set of first-order differential equations in matrix form, which is exactly the form that computers simulate, that modern control and estimation methods use (ROB-300 and ROB-310), and that extends naturally to many joints. This final lesson develops state-space models from the differential equations of earlier lessons, connects them to transfer functions through eigenvalues, discretizes them for implementation, and then pulls the whole course together in a capstone: modeling, analyzing, sampling and filtering a motor-driven robot joint.

State variables and the state-space form

The state of a system is the smallest set of variables that, together with future inputs, determines all future behavior. For a mass-spring-damper, position and velocity form the state: knowing both now, and the force from now on, fixes the motion forever. In general the number of state variables equals the order of the system, which is the number of independent energy stores: each inertia, spring, capacitor or inductor contributes one. A linear time-invariant system in state-space form is

where x is the state vector, u the input vector, y the output vector, and A, B, C and D constant matrices. A describes how the state evolves on its own, B how inputs push it, C which combinations are measured, and D any direct feedthrough from input to output, which is zero for most physical plants. For the mass-spring-damper with state \( \mathbf{x} = [x, \dot{x}]^T \), input force F and measured position,

The recipe is general: write each n-th order equation as n first-order equations by naming the derivatives as new states, then collect the coefficients into matrices.

From state space to transfer function and back

Taking Laplace transforms with zero initial conditions, \( s\mathbf{X} = A\mathbf{X} + B\mathbf{U} \), so \( \mathbf{X} = (sI - A)^{-1}B\mathbf{U} \) and

The inverse \( (sI - A)^{-1} \) has the determinant \( \det(sI - A) \) as its denominator, so the poles of the transfer function are the roots of the characteristic polynomial \( \det(sI - A) = 0 \): the eigenvalues of A. Stability is therefore an eigenvalue question: the system is stable if all eigenvalues of A have negative real parts, or, in discrete time, if all eigenvalues of the discrete A lie inside the unit circle. Going the other way, any transfer function can be written in state-space form, but not uniquely: changing the choice of state variables by any invertible transformation gives a different A, B and C with the same input-output behavior and the same eigenvalues. Standard choices, such as the controllable canonical form, read the matrices directly off the transfer-function coefficients.

Solving and simulating state equations

The state equation has an explicit solution. With no input, \( \mathbf{x}(t) = e^{At}\mathbf{x}(0) \), where the matrix exponential \( e^{At} = I + At + (At)^2/2! + \cdots \) generalizes the scalar exponential of first-order systems. With an input, the solution adds a convolution of the input with \( e^{At}B \), the matrix version of lesson 3. In practice we rarely evaluate these formulas by hand. Instead we discretize. If the input is held constant over each sample period T, as it is behind a zero-order hold (lesson 9), the exact discrete model is

This is the zero-order-hold discretization of lesson 10 in matrix form: each eigenvalue \( \lambda \) of A becomes an eigenvalue \( e^{\lambda T} \) of \( A_d \). The discrete model is what runs inside a simulator, a model-predictive controller or a Kalman filter. A convenient trick computes both matrices at once from a single matrix exponential of an augmented matrix, as the code below does. Forward Euler, \( A_d \approx I + AT \), \( B_d \approx BT \), is simpler but, as lesson 10 warned, can be inaccurate or unstable for stiff systems, such as a motor whose electrical time constant is much shorter than its mechanical one.

import numpy as np
def expm(M, terms=30):                  # matrix exponential by scaling and squaring
    n = max(0, int(np.ceil(np.log2(max(np.abs(M).sum(axis=1).max(), 1e-12)))) + 1)
    Ms = M / 2**n; E = np.eye(M.shape[0]); term = np.eye(M.shape[0])
    for k in range(1, terms):
        term = term @ Ms / k; E = E + term
    for _ in range(n): E = E @ E
    return E
k, R, L, J, b = 0.05, 1.0, 0.5e-3, 2e-4, 1e-5     # motor and joint parameters
A = np.array([[0, 1, 0], [0, -b/J, k/J], [0, -k/L, -R/L]])
B = np.array([[0], [0], [1/L]])
print("eigenvalues of A:", np.round(np.sort(np.linalg.eigvals(A).real), 1))
T = 1e-3
M = np.zeros((4, 4)); M[:3, :3] = A*T; M[:3, 3:] = B*T
E = expm(M); Ad, Bd = E[:3, :3], E[:3, 3:]
print("eigenvalues of Ad (exact):", np.round(np.sort(np.linalg.eigvals(Ad).real), 4))
print("eigenvalues of I + A*T (Euler):", np.round(np.sort(np.linalg.eigvals(np.eye(3) + A*T).real), 4))

eigenvalues of A: [-1987.4   -12.6     0. ]
eigenvalues of Ad (exact): [0.137  0.9874 1.    ]
eigenvalues of I + A*T (Euler): [-0.9874  0.9874  1.    ]

The continuous model has an integrator pole at zero, a mechanical pole and a very fast electrical pole near −2,000 rad/s. Exact discretization maps them all safely inside the unit circle. Forward Euler maps the fast electrical pole to a negative value of magnitude near one, giving a barely stable, sign-alternating mode that does not exist in the real motor. That is exactly the failure lesson 10 predicted for aT near 2.

Capstone: modeling and filtering a motor-driven joint

The final task brings the course together. A small collaborative arm's elbow joint is driven through a 50:1 gearbox by a DC motor; a 4,096-count encoder on the motor shaft measures position, sampled at 1 kHz, and a current sensor is sampled at 20 kHz. The arm's first structural mode is at 35 Hz. We need a model for control design, a sampling and filtering plan, and an estimate of the velocity signal quality. The parameters are \( k = 0.05 \) N·m/A, \( R = 1.0 \) Ω, \( L = 0.5 \) mH, motor-side reflected inertia \( J = 2 \times 10^{-4} \) kg·m² including the link, and negligible friction.

The capstone shows how the topics of this course interlock. Differential equations and state space give the model; transfer functions and Bode plots show where its poles and resonances lie; the sampling theorem and the zero-order hold set the update rates; aliasing dictates how current is measured; and digital filters, budgeted in phase, turn raw encoder counts into usable velocity. The same reasoning applies to every axis of every robot, from a hobby rover to a surgical arm.