The Transition to State Space
The Transition to State Space
Introduction: The Physics Abstraction
When a layman pictures a robotic arm throwing a ball or a drone correcting its pitch in a crosswind, they visualize physical bodies acting in 3-dimensional Euclidean space (the \(X, Y, Z\) coordinates of our physical reality).
But control theorists and kinematic engineers almost never look at the \(X, Y, Z\) space. They work in a wholly abstract, higher-dimensional mathematical realm known as State Space. State space is arguably the most fundamental translation required in control theory. If you do not understand state space, nothing else in the subsequent volumes of Tangent-Space Methods will make sense.
Imagine taking a photograph of a physical system. A photograph captures position, but not motion (you cannot tell if a car in a photo is parked or moving 100 mph). State Space is the mathematical equivalent of the ultimate, complete “photograph.” A single point in State Space contains absolutely everything you need to know about the system to perfectly predict its future (assuming you know the laws of physics dictating it).
Constructing a State Vector
Let us build a state space from the ground up using the simplest dynamical system possible: a 1-Dimensional point mass (like a train on a straight track).
We want to predict what the train will do. To do this, we need to know its position, \(x\). Is \(x\) the “state” of the system?
No. If we only know that the train is at \(x = 100\) meters, we cannot predict where it will be 1 second from now. It could be moving right, moving left, or stationary.
Therefore, to complete our “photograph,” we must also know its velocity, \(v\) (often denoted as the derivative of position, \(\dot{x}\)). If we know \((x, \dot{x})\), we know its exact location and its exact speed.
Do we need to know acceleration (\(\ddot{x}\)) as part of the state? According to Newton’s Second Law: \(F = m\ddot{x}\). Acceleration is not an independent property of the train; it is completely driven by the external forces applied at that instant. Therefore, if we know the forces (the Control Inputs, \(\control\)), we can compute the acceleration. The state itself is fully defined by just position and velocity.
The state vector \(\state\) is the minimum set of mathematical variables completely describing the condition of a dynamic system at a specific point in time. Formally: \(\state(t)\) is a state if the future evolution of the system under a known input \(\control(\cdot)\) depends only on \(\state(t)\) and \(\control(\tau)\) for \(\tau \ge t\)—not on the history of \(\state(\tau)\) for \(\tau < t\). This is the Markov property and is what distinguishes a true state from a mere observation.
For our train, it is a 2-dimensional vector: \[\begin{equation} \state = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} \text{position} \\ \text{velocity} \end{bmatrix} = \begin{bmatrix} q \\ \dot{q} \end{bmatrix} \end{equation}\]
Position, Velocity, and Acceleration: What Lives Where?
Here is a question that trips up even advanced students: if the state includes position and velocity, where does acceleration fit? And are all three of these things “the same kind of object,” mathematically?
The answer is no, and the distinction is beautiful.
Position is a point on a surface. Imagine you are standing on the surface of the Earth. Your position is a location—a specific spot on the globe. It is not an arrow, not a direction, not a magnitude. It is just where you are. Mathematically, position lives on a manifold—a potentially curved surface. The surface of the Earth is curved. The set of all possible configurations of a robot arm (its joint angles) forms a curved surface too. Position is a point on that surface.
Velocity is an arrow at a point. Now imagine you are walking north at 3 mph. Your velocity is a direction and a speed—an arrow pinned to your current location, pointing north, with length proportional to 3 mph. This arrow lives in the tangent space at your position. It does not live on the globe itself (you cannot “add” two locations on a globe in any meaningful way), but you can add two velocity arrows at the same point. If the wind pushes you east at 1 mph while you walk north at 3 mph, your combined velocity is the vector sum—an arrow pointing slightly east of north. This vector addition works because the tangent space is a genuine vector space.
Acceleration is an arrow that tells the velocity arrow how to change. It too lives in the tangent space (or more precisely, in the tangent space of the tangent bundle, but for systems in \(\mathbb{R}^n\), we can think of it as living in the same vector space as velocity). Newton’s second law, \(F = ma\), is the bridge: force (which you can apply) determines acceleration (which changes velocity), which in turn sweeps the position along the manifold.
| Quantity | What it is | Where it lives | Can you add two of them? |
|---|---|---|---|
| Position \(q\) | A point on a surface | The configuration manifold \(\mathcal{Q}\) | No (adding two GPS coordinates is meaningless) |
| Velocity \(\dot{q}\) | An arrow at a point | The tangent space \(T_q\mathcal{Q}\) | Yes (velocities add as vectors) |
| Acceleration \(\ddot{q}\) | How the arrow changes | The tangent space \(T_q\mathcal{Q}\) | Yes (accelerations add as vectors) |
This distinction matters enormously: superposition (adding things together) only works for velocities and accelerations, never for positions. This is why the entire machinery of linear control theory operates in tangent spaces, where addition is valid, rather than on the configuration manifold, where it is not.
Think of it this way. If two friends live at different addresses, “adding” their addresses gives you nonsense. But if both are walking, you can absolutely add their walking speeds. Position is like an address. Velocity is like a walking speed and direction. The tangent space is the space of all possible walking speeds and directions at a given address.
For our train on a track, this distinction is invisible because the track is a straight line (\(\mathbb{R}^1\)), and the tangent space to a straight line looks exactly like the line itself. But for a robot arm with joint angles wrapping around from \(0°\) to \(360°\), the distinction becomes critical. The joint angles live on a circle (a curved manifold), and the velocities live in the tangent line to that circle—a flat, infinite line where addition works perfectly.
Why Acceleration Is Not Part of the State
This is worth repeating because it is so often confused. The state \(\mathbf{x} = [q, \dot{q}]^T\) includes position and velocity but not acceleration. Why?
Because acceleration is not free. It is completely determined by the current state plus the current forces. Newton tells us \(\ddot{q} = F/m\), where \(F\) is whatever force is acting right now. If you know the state (where you are and how fast you are going) and you know the forces (the control inputs plus gravity, friction, etc.), then acceleration is computed, not remembered.
The state captures everything the system “remembers” about its past. Position and velocity are the memories. Acceleration is the consequence of the present moment—the force being applied right now. This is why the state space is \(2n\)-dimensional (n positions + n velocities), not \(3n\)-dimensional.
State Space Representations
Because the state vector \(\state\) contains \(n\) variables, it naturally lives in an \(n\)-dimensional geometry: the State Space, denoted as \(\statespace \in \Reals^n\).
An astonishing realization for beginners is that when a dynamic system (like an airplane or a robotic arm) moves through physical space over time, its entire sprawling motion is represented as a single, unified 1-dimensional curve snaking through \(n\)-dimensional state space. This curve is called the system’s Trajectory \(\traj\).
Standard Form Differential Equations
The immense power of state space is that it allows us to convert complex, higher-order physical differential equations into simple, first-order vector math.
Consider a simple mass-spring-damper system governed by a second-order differential equation: \[\begin{equation} m\ddot{q} + c\dot{q} + kq = u \end{equation}\] where \(q\) is position, \(m\) is mass, \(c\) is damping, \(k\) is stiffness, and \(u\) is our control force.
By defining our state vector as \(\state = [x_1, x_2]^T = [q, \dot{q}]^T\), we can rewrite this physical equation as purely mathematical first-order derivatives:
\[\begin{align} \dot{x}_1 &= x_2 \\ \dot{x}_2 &= \frac{1}{m}(u - cx_2 - kx_1) \end{align}\]
This is called the State Space Representation. It is universally written in nonlinear control literature as:
Every autonomous control system can be written as a first-order vector differential equation mapping current state and current control commands to the velocity of the state vector: \[\begin{equation} \bm{\dot{x}} = \bm{f}(\state, \control, t) \end{equation}\] This vector field \(\bm{f}\) dictates how the system “flows” through the abstract geometry of the state space.
Whether you are controlling a single gear or a 15-DOF musculoskeletal golf swing (as discussed in Volume II), you are always, fundamentally, trying to shape the trajectory \(\state(t)\) by manipulating the vector field \(\bm{f}\) using your control inputs \(\control\).
Summary
State space fundamentally decouples physics from geometry. We no longer treat a system as physical objects bumping into each other in 3D space. We treat it as a single point—a cursor—moving through hyper-dimensional coordinates based on a vector field.
But what defines those coordinates? How do we mathematically decide what the angles and positions of highly complex mechanisms are? For that, we turn to Configuration Space.