Jacobian Matrix Explained: From Velocity Kinematics to Force Control

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Hey friend

The Jacobian matrix is a tool that helps us understand how humanoid robots move and interact with the world. I remember when I first heard about the matrix in a robotics lecture. It sounded really scary.. When I finally understood it everything became clear. I was able to understand how to make the robot move its arm smoothly balance while reaching for something and apply forces.

Today I want to explain the matrix in a simple way using examples of humanoid robots. I will not use much math just what is necessary.

What is the Jacobian matrix?

The Jacobian matrix is a tool that tells us how fast the end of the robot like its hand or foot is moving when we change the velocities. It helps us connect the space, which is the angles or torques at each motor with the task space, which is the actual position and orientation of the hand or foot in the real world.

Think of it like this:

  • You control the joints directly.
  • You care about what the hand or foot’s doing in 3D space.
  • The Jacobian matrix translates between the two.

The Jacobian matrix is like a bridge between the space and the task space. It looks like this:

x˙=J(q)×q˙ẋ = J(q) × q̇

Where ẋ is the velocity of the end-effector J(q) is the matrix and q̇ is the joint velocities.

The Jacobian matrix is really important for humanoid robots. It gives us a lot of power in areas:

  1. Velocity Kinematics: The Jacobian matrix tells us how fast each joint needs to rotate to make the hand move upward at a certain speed.
  2. Inverse Velocity Kinematics: If the Jacobian matrix is invertible we can solve for the required velocities to achieve a desired end-effector velocity.
  3. Torque Control: The Jacobian matrix also works in the direction for forces. It tells us how much torque each joint needs to produce to apply a certain force.
  4. Singularity Detection: When the Jacobian matrix becomes singular the robot loses the ability to move in directions.

Lets look at some examples of humanoid robots that use the matrix:

  • Tesla Optimus uses the Jacobian matrix when reaching for objects while keeping its body balanced.
  • Figure 01 uses the Jacobian matrix for dexterous manipulation tasks like picking up tools or handing objects to humans.
  • Boston Dynamics Atlas uses the full Jacobian matrix for whole-body control allowing it to coordinate its arms, torso and legs together.

When a robot gently grasps an object without crushing it it’s likely that a Jacobian-based force controller is working behind the scenes.

Lets imagine a 2-joint arm. The Jacobian matrix for this arm is a 2×2 matrix. Looks like this:

J=[−l1sin⁡θ1.L2sin⁡(θ1+θ2).−L2sin⁡(θ1+θ2)l1cos⁡θ1+l2cos⁡(θ1+θ2)l2cos⁡(θ1+θ2)]J = \begin{bmatrix} -l_1 \sin\theta_1. L_2 \sin(\theta_1+\theta_2). -L_2 \sin(\theta_1+\theta_2) \\ l_1 \cos\theta_1 + l_2 \cos(\theta_1+\theta_2) & l_2 \cos(\theta_1+\theta_2) \end{bmatrix}

Don’t worry about memorizing it. The important thing is to understand what it represents: it maps velocities to hand velocity.

My personal take on the matrix is that it’s a concept that feels abstract until it suddenly clicks. Then you see it everywhere. It connects everything we’ve talked about far: kinematics, trajectory planning, actuator control and force interaction with the environment.

Once you understand the matrix you start thinking in terms of task space instead of just joint space. That’s when robot control starts feeling more human-like. Modern humanoid robots, like Optimus and Figure don’t just use the Jacobian matrix. They use more advanced versions to handle singularities and prioritize important tasks.

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