Velocity Kinematics

Forward kinematics tells us where the end-effector is. But for real-time control, we also need to know how fast it’s moving and in which direction. This is the domain of velocity kinematics, and its central object is the Jacobian matrix.

From Positions to Velocities

Recall from the previous lesson that forward kinematics gives us a vector function:

\mathbf{x} = f(\mathbf{q})

where $\mathbf{q} = [\theta_1, \theta_2, \ldots, \theta_n]^T$ is the joint angle vector and $\mathbf{x}$ is the end-effector pose. Differentiating with respect to time:

\dot{\mathbf{x}} = \frac{\partial f}{\partial \mathbf{q}} \dot{\mathbf{q}} = J(\mathbf{q}) \dot{\mathbf{q}}

The matrix $J(\mathbf{q})$ is the Jacobian — it linearly maps joint velocities $\dot{\mathbf{q}}$ to end-effector velocities $\dot{\mathbf{x}}$ .

The Jacobian Is a Matrix of Partial Derivatives

Each entry of the Jacobian is a partial derivative:

J_{ij} = \frac{\partial x_i}{\partial q_j}

Row $i$ tells how end-effector component $x_i$ responds to each joint. Column $j$ tells what velocity the end-effector gets when only joint $j$ moves.

The 2-Link Planar Arm Jacobian

For the 2R arm from the previous lesson:

x = L_1\cos\theta_1 + L_2\cos(\theta_1 + \theta_2)

y = L_1\sin\theta_1 + L_2\sin(\theta_1 + \theta_2)

Taking partial derivatives:

J = \begin{bmatrix} \frac{\partial x}{\partial \theta_1} & \frac{\partial x}{\partial \theta_2} \\[6pt] \frac{\partial y}{\partial \theta_1} & \frac{\partial y}{\partial \theta_2} \end{bmatrix} = \begin{bmatrix} -L_1 s_1 - L_2 s_{12} & -L_2 s_{12} \\ L_1 c_1 + L_2 c_{12} & L_2 c_{12} \end{bmatrix}

where $s_1 = \sin\theta_1$ , $c_1 = \cos\theta_1$ , $s_{12} = \sin(\theta_1 + \theta_2)$ , $c_{12} = \cos(\theta_1 + \theta_2)$ .

Reading the Jacobian

Column 1 gives the end-effector velocity when only joint 1 rotates at unit speed ( $\dot\theta_1 = 1, \dot\theta_2 = 0$ ). This is the velocity from the entire arm swinging around the base.
Column 2 gives the velocity when only joint 2 rotates ( $\dot\theta_1 = 0, \dot\theta_2 = 1$ ). This is just the second link swinging around joint 2.

The end-effector velocity is a linear combination:

\begin{bmatrix} \dot{x} \\ \dot{y} \end{bmatrix} = \dot\theta_1 \cdot \mathbf{J}_1 + \dot\theta_2 \cdot \mathbf{J}_2

where $\mathbf{J}_1$ and $\mathbf{J}_2$ are the columns of $J$ .

Velocity Computation Robotics Application

At $\theta_1 = 45°, \theta_2 = 90°$ with $L_1 = 2, L_2 = 1.5$ :

s_1 = 0.707, \quad c_1 = 0.707, \quad s_{12} = \sin(135°) = 0.707, \quad c_{12} = \cos(135°) = -0.707

J = \begin{bmatrix} -2(0.707) - 1.5(0.707) & -1.5(0.707) \\ 2(0.707) + 1.5(-0.707) & 1.5(-0.707) \end{bmatrix} = \begin{bmatrix} -2.475 & -1.061 \\ 0.354 & -1.061 \end{bmatrix}

If joint 1 rotates at 1 rad/s and joint 2 is stationary:

\dot{\mathbf{x}} = J \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} -2.475 \\ 0.354 \end{bmatrix} \text{ m/s}

The end-effector moves mostly in the $-x$ direction — it’s swinging leftward.

Singularities

A configuration is singular when the Jacobian loses rank — that is, when $\det(J) = 0$ . At a singularity, there exist directions in which the end-effector cannot move regardless of joint velocities.

For the 2R arm:

\det(J) = L_1 L_2 \sin\theta_2

This is zero when $\theta_2 = 0$ or $\theta_2 = \pi$ :

$\theta_2 = 0$ (fully extended): The arm is a straight line. It can swing sideways but cannot extend or retract radially.
$\theta_2 = \pi$ (fully folded): The arm folds back on itself. Same problem — no radial motion.

Singularities in Practice

Near a singularity:

Some end-effector velocities require infinite joint velocities (the robot can’t keep up)
The inverse Jacobian $J^{-1}$ blows up — computed joint commands become unreasonably large
The robot may exhibit jerky, unstable motion
Force control becomes problematic: the robot can exert very large forces in certain directions with tiny joint torques

Industrial robots have singularity-avoidance algorithms that detect when the robot approaches a singular configuration and modify the trajectory accordingly.

Singularities of 6-DOF Arms

For a 6-DOF arm, singularities are more complex. Common cases include:

Shoulder singularity: The wrist center lies on the $z_0$ axis (directly above or below the base)
Elbow singularity: The arm is fully extended or folded (like the 2R case)
Wrist singularity: Two of the wrist axes align (similar to gimbal lock)

The Manipulability Ellipsoid

The manipulability ellipsoid visualizes how “capable” the robot is at a given configuration. It answers: if the joints can produce unit-speed velocities, what is the set of achievable end-effector velocities?

Mathematically, the set $\{\dot{\mathbf{x}} : \|\dot{\mathbf{q}}\| \leq 1\}$ forms an ellipsoid in task space. This ellipsoid is characterized by $J J^T$ :

\dot{\mathbf{x}}^T (J J^T)^{-1} \dot{\mathbf{x}} \leq 1

The eigenvectors of $J J^T$ give the principal directions of the ellipsoid, and the square roots of the eigenvalues ( $\sigma_1, \sigma_2, \ldots$ ) are the singular values — the semi-axis lengths.

Connection to Eigenvalues (Module 4)

The manipulability ellipsoid is a direct application of eigenanalysis from Module 4. The eigenvalues of $J J^T$ determine the ellipsoid shape:

Large eigenvalue → the robot can move fast in that direction
Small eigenvalue → the robot struggles in that direction
Zero eigenvalue → singularity — motion is impossible in that direction

The condition number $\kappa = \sigma_1 / \sigma_2$ measures isotropy. When $\kappa = 1$ , the ellipsoid is a circle (uniform capability). As $\kappa \to \infty$ , the robot approaches a singularity.

Yoshikawa’s Manipulability Measure

A single scalar summarizing the robot’s overall dexterity:

w = \sqrt{\det(J J^T)} = \sigma_1 \cdot \sigma_2 \cdots \sigma_m

$w = 0$ at singularities
$w$ is maximized at the most “dexterous” configurations
For the 2R arm: $w = |L_1 L_2 \sin\theta_2|$ , which is maximized at $\theta_2 = \pm 90°$

Optimal Configuration Robotics Application

For a 2R arm with $L_1 = 2, L_2 = 1.5$ :

At $\theta_2 = 90°$ : $w = 2 \times 1.5 \times \sin(90°) = 3.0$ (maximum)
At $\theta_2 = 45°$ : $w = 2 \times 1.5 \times \sin(45°) \approx 2.12$
At $\theta_2 = 10°$ : $w = 2 \times 1.5 \times \sin(10°) \approx 0.52$ (poor)
At $\theta_2 = 0°$ : $w = 0$ (singular)

The best configuration is the right-angle elbow ( $\theta_2 = 90°$ ), where the manipulability ellipsoid is closest to a circle. You can verify this with the interactive visualization below.

The 6-DOF Jacobian

For a general 6-DOF arm, the Jacobian is a 6×6 matrix mapping 6 joint velocities to the 6-dimensional end-effector velocity (3 linear + 3 angular):

\begin{bmatrix} \mathbf{v} \\ \boldsymbol{\omega} \end{bmatrix} = J(\mathbf{q}) \dot{\mathbf{q}}

where $\mathbf{v} = [\dot{x}, \dot{y}, \dot{z}]^T$ is linear velocity and $\boldsymbol{\omega} = [\omega_x, \omega_y, \omega_z]^T$ is angular velocity. Each column of $J$ has two parts:

\mathbf{J}_i = \begin{bmatrix} \mathbf{J}_{v,i} \\ \mathbf{J}_{\omega,i} \end{bmatrix}

For revolute joint $i$ , these are computed from the forward kinematics:

\mathbf{J}_{v,i} = \hat{z}_{i-1} \times (\mathbf{p}_n - \mathbf{p}_{i-1}), \quad \mathbf{J}_{\omega,i} = \hat{z}_{i-1}

where $\hat{z}_{i-1}$ is the rotation axis of joint $i$ and $\mathbf{p}_{i-1}$ is the origin of frame $\{i-1\}$ . The linear velocity contribution is the cross product of the rotation axis with the vector from the joint to the end-effector — this is the formula for the velocity of a point on a rotating body.

Resolved-Rate Motion Control

The Jacobian enables a powerful control strategy: instead of planning in joint space, plan a desired end-effector velocity $\dot{\mathbf{x}}_d$ and compute the required joint velocities:

\dot{\mathbf{q}} = J^{-1}(\mathbf{q}) \dot{\mathbf{x}}_d

This is resolved-rate control. In practice, we use the pseudoinverse $J^+$ for robustness:

\dot{\mathbf{q}} = J^+(\mathbf{q}) \dot{\mathbf{x}}_d

Near singularities, the damped least-squares (DLS) inverse is preferred:

\dot{\mathbf{q}} = J^T (J J^T + \lambda^2 I)^{-1} \dot{\mathbf{x}}_d

where $\lambda$ is a damping factor that trades off tracking accuracy for numerical stability. This connects directly to the matrix inverse and condition number concepts from Module 2.

Static Force Analysis

The Jacobian also relates joint torques $\boldsymbol{\tau}$ to end-effector forces $\mathbf{f}$ through the principle of virtual work:

\boldsymbol{\tau} = J^T \mathbf{f}

The transpose of the Jacobian maps task-space forces to joint-space torques. This duality between velocity and force is a fundamental result of Lagrangian mechanics.

Interactive Visualization

Explore the Jacobian for a 2-link planar arm. Adjust joint angles and velocities to see how the Jacobian maps joint motion to end-effector velocity. The manipulability ellipsoid shows the robot’s directional capability at each configuration.

Controls

Configuration

θ₁ = 45°

-3.1415926535897933.141592653589793

θ₂ = 60°

-3.1415926535897933.141592653589793

Joint Velocities

dθ₁/dt = 1.0 rad/s

-22

dθ₂/dt = 0.0 rad/s

-22

Show manipulability ellipsoid

Show velocity vector

Jacobian J(θ)

[	-2.86	-1.45	]
[	1.03	-0.39	]

Analysis

det(J) = 2.598

w (manipulability) = 2.598

σ₁ = 3.298σ₂ = 0.788

condition # = 4.2

v_ee = (-2.86, 1.03)

|v_ee| = 3.041

Try this: Set "Right Angle" for a well-conditioned configuration (round ellipsoid). Then click "Singular" (θ₂ → 0) and watch the ellipsoid collapse to a line — the robot loses the ability to move radially. Try different joint velocity inputs to see how the Jacobian maps them to end-effector velocity.

Practice Problems

For a 2R arm with $L_1 = L_2 = 1$ , compute the Jacobian at $\theta_1 = 0, \theta_2 = 90°$ . What is $\det(J)$ ?
At the configuration from Problem 1, if $\dot\theta_1 = 1$ rad/s and $\dot\theta_2 = -1$ rad/s, what is the end-effector velocity?
Find all singular configurations for a 2R arm. Describe geometrically what happens at each.
A 2R arm has $L_1 = 3, L_2 = 2$ . At what value of $\theta_2$ is the manipulability measure $w$ maximized? What is $w_{\max}$ ?
For the 6-DOF Jacobian column formula $\mathbf{J}_{v,i} = \hat{z}_{i-1} \times (\mathbf{p}_n - \mathbf{p}_{i-1})$ , explain why this gives the correct linear velocity contribution from joint $i$ .

Answers

At $\theta_1 = 0, \theta_2 = 90°$ : $s_1 = 0, c_1 = 1, s_{12} = 1, c_{12} = 0$ . $J = \begin{bmatrix} -0 - 1 & -1 \\ 1 + 0 & 0 \end{bmatrix} = \begin{bmatrix} -1 & -1 \\ 1 & 0 \end{bmatrix}$ . $\det(J) = (-1)(0) - (-1)(1) = 1 = L_1 L_2 \sin(90°)$ .
$\dot{\mathbf{x}} = J \begin{bmatrix} 1 \\ -1 \end{bmatrix} = \begin{bmatrix} -1+1 \\ 1+0 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$ m/s. The end-effector moves straight up.
Singular when $\sin\theta_2 = 0$ , i.e., $\theta_2 = 0$ (arm fully extended — cannot move radially inward/outward) or $\theta_2 = \pi$ (arm fully folded — same issue, also the end-effector is at a distance of $|L_1 - L_2|$ from the base).
$w = L_1 L_2 |\sin\theta_2| = 6|\sin\theta_2|$ . Maximized at $\theta_2 = \pm 90°$ with $w_{\max} = 6$ .
Joint $i$ rotates frame $\{i-1\}$ about $\hat{z}_{i-1}$ . The end-effector is at distance $\|\mathbf{p}_n - \mathbf{p}_{i-1}\|$ from the joint. The velocity of a point at distance $\mathbf{r}$ from a rotation axis $\hat{\omega}$ rotating at rate $\dot\theta$ is $\mathbf{v} = \hat{\omega} \times \mathbf{r} \cdot \dot\theta$ . With $\hat{\omega} = \hat{z}_{i-1}$ and $\mathbf{r} = \mathbf{p}_n - \mathbf{p}_{i-1}$ , this is exactly $\mathbf{J}_{v,i} \cdot \dot\theta_i$ .

Key Takeaways

The Jacobian $J(\mathbf{q})$ maps joint velocities to end-effector velocities: $\dot{\mathbf{x}} = J \dot{\mathbf{q}}$
Each Jacobian entry is a partial derivative: $J_{ij} = \partial x_i / \partial q_j$
Singularities occur when $\det(J) = 0$ — some end-effector motions become impossible
The manipulability ellipsoid (from the eigenvalues of $J J^T$ ) visualizes directional capability
The Jacobian transpose maps end-effector forces to joint torques: $\boldsymbol{\tau} = J^T \mathbf{f}$
Resolved-rate control uses $J^{-1}$ (or $J^+$ ) to compute joint velocities for a desired task-space motion

Next Steps

The final lesson brings these tools to bear on path planning — using potential fields, gradients, and trajectory interpolation to move robots through cluttered environments.