Compute Maximum End Effector Velocity Along Path

Let's say I have a 2-link planar robot arm. The motors in both joints have maximum angular speeds that limit how fast they can spin. If I plan a path from point A to point B with a constant end effector velocity, how do I calculate the maximum velocity the end effector can travel with along this path?

I know I can calculate the velocity of the end effector with this equation: $$v=J(q)\dot{q}$$. I could iterate through points along the path plugging in the joint's maximum angular speeds to calculate the maximum velocity at each point. Then I would use the smallest velocity as the "maximum" velocity the end effector can travel at.

Is this the best way to accomplish my goal?

• I have modified my answer as I misinterpreted your method at a first glance. Apr 18 at 20:31

Nope, you cannot simply plug the maximum joint velocities in the differential forward kinematic law, as you need to guarantee that $$\mathbf{v}$$ will always point toward the target: in fact, $$\mathbf{v}$$ is a vector.

As a result, you need to solve the inverse kinematics problem first for each point of the path. This can be done iteratively:

1. Assign a desired $$\mathbf{v}$$ with a given magnitude and direction.
2. Solve for each point $$\dot{\mathbf{q}} = \mathbf{J}^{-1} \cdot \mathbf{v}$$.
3. Check if the components of $$\dot{\mathbf{q}}$$ are within the allowed range of the motors throughout the entire path. If not, reduce the magnitude of $$\mathbf{v}$$ and iterate again.

In practice, you ought to consider that the kinematic singularities may impede the simple solution resorting to the (pseudo)inverse of the Jacobian.

To get around the singularities, one usually resorts to the Damped Least-Squares inversion $$\mathbf{J}^*$$ of the Jacobian $$\mathbf{J}$$, where

$$\mathbf{J}^* = \mathbf{J}^T \left( \mathbf{J}\mathbf{J}^T + k^2\mathbf{I} \right).$$

Then, because of the tuning of $$k$$, you won't have most likely a unique correspondence between the joint and the end-effector velocity, but rather you would need to reason in terms of margins.

To elaborate more on the particular 2D case, I could argue the following:

1. We need first to solve the IK problem and come up with the exact solution $$f \left( q_1, q_2 \right) = \left( x, y \right)^T$$.
2. Then, we need to solve a constrained optimization problem to find the couple $$\left( \dot{q}_1, \dot{q}_2 \right)$$ that maximizes the Cartesian velocity knowing that this velocity is required to be along a given direction and we have bounds for the speed of the motors.

Of course, the above procedure needs to be iterated throughout the points of the entire path anyway, to finally determine which is the lowest Cartesian velocity that can be attained. This will be your final value as it will guarantee that the bounds on the motor speeds won't be violated at any point of the path.

Here below some more details.

Step 1

We know the exact solution of the IK problem: $$\begin{cases} q_1 = \arctan{\frac{y}{x}} + \tan{\frac{a_2\sin{q_2}}{a_1+a_2\cos{q_2}}} \\ q_2 = -\arccos{\frac{x^2 + y^2 -a^2_1 -a^2_2}{2a_1a_2}} \end{cases},$$ where $$a_1$$, $$a_2$$ are the link lengths.

Step 2

The velocity equations are: $$\begin{cases} \dot{x} = -a_1\sin{q_1}\cdot\dot{q_1} -a_2\sin{\left(q_1+q_2\right)} \cdot \left(\dot{q_1}+\dot{q_2}\right) \\ \dot{y} = a_1\cos{q_1}\cdot\dot{q_1} +a_2\cos{\left(q_1+q_2\right)} \cdot \left(\dot{q_1}+\dot{q_2}\right) \end{cases},$$ where $$\left(q_1, q_2\right)$$ are determined in step 1.

The Cartesian velocities $$\left( \dot{x}, \dot{y} \right)$$ cannot be generic as they need to enforce the assigned direction; so, it holds that $$\frac{\dot{y}}{\dot{x}} = \text{known constant} = c$$.

By aggregating all this information, we get that the solution $$\left( \dot{q_1}, \dot{q_2} \right)^*$$ is the outcome of the following constrained optimization problem:

$$\begin{array}{lcr} \left( \dot{q_1}, \dot{q_2} \right)^* = \arg \max\limits_{\dot{q_1},\dot{q_2}} \frac{1}{2}\left( \dot{y}\left(\dot{q_1},\dot{q_2}\right) \right)^2 \\ \text{s.t.} \begin{cases} \frac{\dot{y}}{\dot{x}} \left(\dot{q_1},\dot{q_2}\right) = c \\ -M_1 < \dot{q_1} < M_1 \\ -M_2 < \dot{q_2} < M_2 \end{cases} \end{array},$$ where $$M_1$$ and $$M_2$$ are the motors max speeds.

The 2D case boils down to the maximization of a quadratic cost function subject to a hyperbolic constraint.

Concluding remarks

In the 2-steps procedure, I didn't mention that you would likely need to deal with kinematic singularities depending on the type of path you have in mind. This will certainly add up more difficulties on top of the process so that, I reckon, the fully-iterative routine detailed in the first place tends to be easier to implement, clearer to maintain, and definitely more scalable with respect to the number of DOF.

• So the only way to find the largest possible vector v at a specific point is iteratively? And the only way to find the maximum possible v along a path of constant velocity is also iteratively? Apr 18 at 20:59
• Given a path, there ain't two nested iterations, but only one at the level of the magnitude $v$. Anyway, my point was general and works for all numbers of DOF. In your case, if it's 2D only, there exists an exact solution to the inverse kinematics $\mathbf{J}^{-1}$, so you don't even need to iterate, probably. Apr 18 at 21:43
• Uhm, maybe it can be a bit more complicated than my oversimplification in my comment above. I could come back to this later. Apr 18 at 21:52
• Ok thank you for your help! Yes I eventually want to scale this concept up to a 6DOF arm but I want to learn it on a 2DOF arm to start. Apr 18 at 23:11
• I've tried to make one step forward and summarized above what I had in mind. Apr 19 at 11:27