Partial Derivatives & Jacobians for Kinematics & State Estimation

1. Linearizing the Non-Linear World

Most realistic models in robotics are non-linear. For example, the relationship between a robot's wheel velocities and its change in global pose $(x,y,\theta )$ involves trigonometric functions. Similarly, a radar sensor might measure range and bearing, which relate to Cartesian coordinates through non-linear equations.

If our belief is a Gaussian and we pass it through a non-linear function, the result is no longer a Gaussian. This breaks the elegant mathematical machinery we want to use. The solution is not to use the exact non-linear function, but to use a linear approximation of it at the current point of interest (our current best state estimate, $\mathbf{\mu }$).

For a single-variable function $y=f(x)$, you know that the best linear approximation around a point $x_0$ is given by the tangent line, whose slope is the derivative $f^{\prime }(x_0)$. The Jacobian is the generalization of the derivative to vector-valued functions of multiple variables.

As the video explains, for a function $\mathbf{f}:{\mathbb{R}}^n\to {\mathbb{R}}^m$ that maps an n-dimensional input vector $\mathbf{x}$ to an m-dimensional output vector $\mathbf{y}$, the Jacobian matrix $\mathbf{J}$ is an $m\times n$ matrix. Each element $J_{ij}$ is the partial derivative $\frac{\partial f_i}{\partial x_j}$, which measures how the i-th output component changes with a small change in the j-th input component.

2. The Jacobian and the Taylor Series Expansion

The reason the Jacobian provides the best linear approximation is rooted in the multivariate Taylor series expansion. Just as a 1D function can be approximated by $f(x)\approx f(x_0)+f^{\prime }(x_0)(x-x_0)$, a multivariate function can be approximated as:

Here, ${\mathbf{J}}_{\mathbf{f}}({\mathbf{x}}_0)$ is the Jacobian of $\mathbf{f}$ evaluated at the point ${\mathbf{x}}_0$. This equation is the foundation for the Extended Kalman Filter (EKF), a topic we will dive into in a future module.

The key takeaway is that the Jacobian matrix allows us to treat a non-linear function as a linear one locally, around a specific operating point. This is exactly what we need to propagate the mean and covariance of our belief through a non-linear system model.

3. The Jacobian in Kinematics: Relating Joint and End-Effector Velocity

Now, let's switch gears to a different, but equally important, application of the Jacobian: robot kinematics. Consider a robotic arm. The position of its end-effector (the "hand") is a non-linear function of its joint angles. This is called forward kinematics.

A central question in robot control is: if I move my joints with certain velocities ($\dot{\mathbf{q}}$), what will be the resulting linear and angular velocity of my end-effector (${\mathbf{v}}_{ee}$)? This relationship, called differential kinematics, is given by the Jacobian.

The relationship is surprisingly linear:

Where:

• ${\mathbf{v}}_{ee}$ is the velocity vector of the end-effector.
• $\dot{\mathbf{q}}$ is the vector of joint velocities.
• $\mathbf{J}(\mathbf{q})$ is the Robot Jacobian, which is a function of the current joint configuration $\mathbf{q}$.

Let's see how this is derived.

4. Computing the Jacobian for a Kinematic Chain

Let's solidify this by walking through the computation. For the 2-link planar arm with link lengths $L_1$ and $L_2$, the forward kinematics equations for the end-effector position $(x_e,y_e)$ are:

Here, our function is ${\mathbf{p}}_e=\mathbf{f}(\mathbf{q})$, where ${\mathbf{p}}_e=[x_e,y_e{]}^T$ and $\mathbf{q}=[{\theta }_1,{\theta }_2{]}^T$. The Jacobian is a 2x2 matrix:

Let's compute the elements. Using the chain rule for derivatives:

• $\frac{\partial x_e}{\partial {\theta }_1}=-L_1\sin ({\theta }_1)-L_2\sin ({\theta }_1+{\theta }_2)$
• $\frac{\partial x_e}{\partial {\theta }_2}=-L_2\sin ({\theta }_1+{\theta }_2)$
• $\frac{\partial y_e}{\partial {\theta }_1}=L_1\cos ({\theta }_1)+L_2\cos ({\theta }_1+{\theta }_2)$
• $\frac{\partial y_e}{\partial {\theta }_2}=L_2\cos ({\theta }_1+{\theta }_2)$

You can check your work by consulting a formal robotics text.

The MIT resource also introduces the important concept of singularity, which occurs when the Jacobian matrix loses rank (its determinant is zero). At a singular configuration (e.g., when the arm is fully stretched out or folded back on itself), there are directions the end-effector cannot move, no matter how the joints move. This is a critical consideration in robot design and path planning.

Conclusion

In this lesson, we've explored the Jacobian matrix, a fundamental tool with a powerful dual identity in robotics. You now understand that this single mathematical object—a matrix of partial derivatives—plays two distinct but equally vital roles.

Key Takeaways:

• The Jacobian matrix is the generalization of the derivative for vector-valued functions. It represents the best local linear approximation of a non-linear function around a specific point.
• In non-linear state estimation, the Jacobian is used to linearize system models (like motion or sensor models). This allows us to propagate a Gaussian belief (a mean and covariance) through the model, forming the basis of algorithms like the Extended Kalman Filter.
• In kinematics, the Jacobian relates the velocities of a robot's joints to the resulting velocity of its end-effector. This is essential for controlling the motion of robotic arms and other articulated mechanisms.

You've now assembled the core mathematical building blocks for the course: homogeneous transforms for poses, Gaussian distributions for uncertainty, and Jacobians for handling non-linear dynamics.

In our next lesson, we will begin to assemble these tools. We will start with the foundational principle of probabilistic inference, Bayes' Rule, and see how it provides a formal recipe for updating our belief in light of new information. This will pave the way for implementing our first full-fledged filter.