Perception, Planning, and Control

© Russ Tedrake, 2020-2022

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**Note:** These are working notes used for a course being taught
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There are a few more essential skills that we need in our toolbox. In this chapter, we will explore some of the powerful methods of kinematic trajectory motion planning.

I'm actually almost proud of making it this far into the notes without
covering this topic yet. Writing a relatively simple script for the pose of
the gripper, like we did in the bin picking chapter, really can solve a lot of
interesting problems. But there are a number of reasons that we might want a
more automated solution:

- When the environment becomes more cluttered, it is harder to write such a simple solution, and we might have to worry about collisions between the arm and the environment as well as the gripper and the environment.
- If we are doing "mobile manipulation" -- our robotic arms are attached to a mobile base -- then the robot might have to operate in many different environments. Even if the workspace is not geometrically complicated, it might still be different enough each time we reach that it requires automated (but possibly still simple) planning.
- If the robot is operating in a simple known environment all day long, then it probably makes sense to optimize the trajectories that it is executing; we can often speed up the manipulation process significantly.

We'll focus on the problem of moving an arm through space in this chapter, because that is the classical and very important motivation. But I need to cover this now for another reason, too: it will also soon help us think about programming a more dexterous hand.

I do need to make one important caveat. Despite having done some work in this field myself, I actually really dislike the problem formulation of collision-free motion planning. I think that on the whole, robots are too afraid of bumping into the world (because things still go wrong when they do). I don't think humans are solving these complex geometric problems every time we reach... even when we are reaching in dense clutter (I actually suspect that we are very bad at it). I would much rather see robots that perform well even with very coarse / approximate plans for moving through a cluttered environment, that are not afraid to make incidental contacts, and that can still accomplish the task when they do!

The goal of this chapter is to solve for motion trajectories. But I would argue that if you really understand how to solve inverse kinematics, then you've got most of what you need to plan trajectories.

We know that that the forward kinematics give us a (nonlinear) mapping from joint angles to e.g. the pose of the gripper: $X^G = f_{kin}(q)$. So, naturally, the problem of inverse kinematics (IK) is about solving for the inverse map, $q = f^{-1}_{kin}(X^G).$ Indeed, that is perhaps the most common and classical question studied in inverse kinematics. But I want you to think of inverse kinematics as a much richer topic than that.

For example, when we were evaluating grasp candidates for bin picking, we had only a soft preference on the orientation of the hand relative to some antipodal grasp. In that case, a full 6 DOF pose would have been an overly constrained specification. And often we have many constraints on the kinematics: some in joint space (like joint limits) and others in Cartesian space (like non-penetration constraints). So really, inverse kinematics is about solving for joint angles in a very rich landscape of objectives and constraints.

With its obvious importance in robotics, you probably won't be
surprised to hear that there is an extensive literature on inverse
kinematics. But you may be surprised at how extensive and complete the
solutions can get. The forward kinematics, $f_{kin}$, is a nonlinear
function in general, but it is a very structured one. In fact, with rare
exceptions (like if your robot has a helical
joint, aka screw joint), the equations governing the valid Cartesian
positions of our robots are actually *polynomial*. "But wait! What
about all of those sines and cosines in my kinematic equations?" you say.
The trigonometric terms come when we want to relate joint angles with
Cartesian coordinates. In $\Re^3$, for two points, $A$ and $B$, on the
same rigid body, the (squared) distance between them, $\|p^A - p^B\|^2,$
is a constant. And a joint is just a polynomial constraint between
positions on adjoining bodies, e.g. that they occupy the same point in
Cartesian space. See

Understanding the solutions to polynomial equations is the subject of
algebraic geometry. There is a deep literature on kinematics theory, on
symbolic algorithms, and on numerical algorithms. For even very complex
kinematic topologies, such as four-bar
linkages and Stewart-Gough
platforms, we can count the number of solutions, and/or understand
the continuous manifold of solutions. For instance,

While the algebraic-geometry methods are mostly targeted for offline
global analysis, they are not designed for fast real-time inverse
kinematics solutions needed in a control loop. The most popular tool
these days for real-time inverse kinematics for six- or seven-DOF
manipulators is a tool called "IKFast", described in Section 4.1
of

These explicit solutions are important to understand because they provide deep insight into the equations, and because they can be fast enough to use inside a more sophisticated solution approach. But the solutions don't provide the rich specification I advocated for above; in particular, they break down once we have inequality constraints instead of equality constraints. For those richer specifications, we will turn to optimization.

We have already
discussed the idea of solving *differential* inverse kinematics
as an optimization problem. In that workflow, we started by using the
pseudo-inverse of the kinematic Jacobian, but then graduated to thinking
about the least-squares formulation of the inverse problem. The more
general least-squares setting, we could add additional costs and
constraints that would protect us from (nearly) singular Jacobians and
could take into account additional constraints from joint limits, joint
velocity limits, etc. We could even add collision avoidance constraints.
Some of these constraints are quite nonlinear / nonconvex functions of
the configuration $q$, but in the differential kinematics setting we were
only seeking to find a small change $\Delta q$ around the nominal
configuration, so it was quite reasonable to make linear/convex
approximations of these nonlinear/nonconvex constraints.

Now we will consider the full formulation, where we try to solve the nonlinear / nonconvex optimization directly, without any constraints on only making a small change to an initial $q$. This is a much harder problem computationally. Using powerful nonlinear optimization solvers like SNOPT, we are often able to solve the problems, even at interactive rates (the example below is quite fun). But there are no guarantees. It could be that a solution exists even if the solver returns "infeasible".

Of course, the differential IK problem and the full IK problem are closely related. In fact, you can think about the differential IK algorithm as doing one step of (projected) gradient descent or one-step of SQP, for the full nonlinear problem.

Drake provides a nice InverseKinematics
class that makes it easy to assemble many of the standard
kinematic/multibody constraints into a
`MathematicalProgram`

. Take a minute to look at the
constraints that are offered. You can add constraints on the relative
position and/or orientation on two bodies, or that two bodies are more
than some minimal distance apart (e.g. for non-penetration) or closer
than some distance, and more. This is the way that I want you to think
about the IK problem; it is an inverse problem, but one with a
potentially very rich set of costs and constraints.

Despite the nonconvexity of the problem and nontrivial computational cost of evaluating the constraints, we can often solve it at interactive rates. I've assembled a few examples of this in the chapter notebook:

In the first version, I've added sliders to let you control the desired pose of the end-effector. This is the simple version of the IK problem, amenable to more explicit solutions, but we nevertheless solve it with our full nonlinear optimization IK engine (and it does include joint limit constraints). This demo won't look too different from the very first example in the notes, where you used teleop to command the robot to pick up the red brick. In fact, differential IK offers a fine solution to this problem, too.

In the second example, I've tried to highlight the differences between the nonlinear IK problem and the differential IK problem by adding an obstacle directly in front of the robot. Because both our differential IK and IK formulations are able to consume the collision-avoidance constraints, both solutions will try to prevent you from crashing the arm into the post. But if you move the target end-effector position from one side of the post to the other, the full IK solver can switch over to a new solution with the arm on the other side of the post, but the differential IK will never be able to make that leap (it will stay on the first side of the post, refusing to allow a collision).

With great power comes great responsibility. The inverse kinematics toolbox allows you to formulate complex optimizations, but your success with solving them will depend partially on how thoughtful you are about choosing your costs and constraints. My basic advice is this:

- Try to keep the objective (costs) simple; I typically only use the "joint-centering" quadratic cost on $q$. Putting terms that should be constraints into the cost as penalties leads to lots of cost-function tuning, which can be a nasty business.
- Write minimal constraints. You want the set of feasible configurations to be as big as possible. For instance, if you don't need to fully constrain the orientation of the gripper, then don't do it.

Let's use IK to grasp a cylinder (call it a hand rail). Suppose it doesn't matter where along the cylinder we grasp, nor the orientation at which we grasp it. Then we should write the IK problem using only the minimal version of those constraints.

In the notebook, I've coded up one version of this. I've put the cylinder's pose on the sliders now, so you can move it around the workspace, and watch how the IK solver decides to position the robot. In particular, if you move the cylinder in $\pm y$, you'll see that the robot doesn't try to follow... until the hand gets to the end of the cylinder. Very nice!

One could imagine multiple ways to implement that constraint. Here's how I did it:

```
ik.AddPositionConstraint(
frameB=gripper_frame, p_BQ=[0, 0.1, -0.02],
frameA=cylinder_frame, p_AQ_lower=[0, 0, -0.5], p_AQ_upper=[0, 0, 0.5])
ik.AddPositionConstraint(
frameB=gripper_frame, p_BQ=[0, 0.1, 0.02],
frameA=cylinder_frame, p_AQ_lower=[0, 0, -0.5], p_AQ_upper=[0, 0, 0.5])
```

In words, I've defined two points in the gripper frame; in the notation
of the `AddPositionConstraint`

method they are ${}^Bp^{Q}$.
Recall the gripper frame is such
that $[0, .1, 0]$ is right between the two gripper pads; you should
take a moment to make sure you understand where $[0,.1,-0.02]$ and
$[0,.1,0.02]$ are. Our constraints require that both of those points
should lie exactly on the center line segment of the cylinder. This
was a compact way for me to leave the orientation around the cylinder
axis as unconstrained, and capture the cylinder position constraints
all quite nicely.
We've provided a rich language of constraints for specifying IK problems, including many which involve the kinematics of the robot and the geometry of the robot and the world (e.g., the minimum-distance constraints). Let's take a moment to appreciate the geometric puzzle that we are asking the optimizer to solve.

Let's return to the example of the iiwa reaching into the shelf. This IK problem has two major constraints: 1) we want the center of the target sphere to be in the center of the gripper, and 2) we want the arm to avoid collisions with the shelves. In order to plot these constraints, I've frozen three of the joints on the iiwa, leaving only the three corresponding motion in the $x-z$ plane.

To visualize the constraints, I've sampled a dense grid in the three joint angles of the planarized iiwa, assigning each grid element to 1 if the constraint is satisfied or 0 otherwise, then run a marching cubes algorithm to extract an approximation of the true 3D geometry of this constraint in the configuration space. The "grasp the sphere" constraint produces the nice green geometry I've pictured above on the right; it is clipped by the joint limits. The collision-avoidance constraint, on the other hand, is quite a bit more complicated. To see that, you'd better open up this 3D visualization so you can navigate around it yourself. Scary!

Zoomed in. The global optimization problem. Nonlinear optimizers like SNOPT can be pretty amazing sometimes!

Global IK.

When should we use IK vs Differential IK? IK solves a more global problem, but is not guaranteed to succeed. It is also not guaranteed to make small changes to $q$ as you make small changes in the cost/constraints; so you might end up sending large $\Delta q$ commands to your robot. Use IK when you need to solve the more global problem, but then produce the actual $q$ trajectories with trajectory optimization, etc. Or use differential IK if you are able to design smooth commands in end-effector space and simply track them in joint space.

In our previous approach, we designed (simple) trajectories in the end-effector coordinates, then used differential IK to get the joint velocity commands. Now we will switch to design the motions directly in joint coordinates, and just add (forward) kinematics constraints.

Piecewise polynomial trajectories

B-spline trajectories

B-spline kinematic trajectory optimization.

CHOMP

The Open Motion Planning Library. We have our own implementations in Drake that are optimized for our collision engines.

Task and motion planning, graphs of convex sets, ...

For this exercise, you will implement a optimization-based inverse kinematics solver to open a cupboard door. You will work exclusively in . You will be asked to complete the following steps:

- Write down the constraints on the IK problem of grabbing a cupboard handle.
- Formalize the IK problem as an instance of optimization.
- Implement the optimization problem using MathematicalProgram.

For this exercise, you will implement and analyze the RRT algorithm introduced in class. You will work exclusively in . You will be asked to complete the following steps:

- Implement the RRT algorithm for the Kuka arm.
- Answer questions regarding its properties.

Due to the random process by which nodes are generated, the paths output by RRT can often look somewhat jerky (the "RRT dance" is the favorite dance move of many roboticists). There are many strategies to improve the quality of the paths and in this question we'll explore two. For the sake of this problem, we'll assume path quality refers to path length, i.e. that the goal is to find the shortest possible path, where distance is measured as Euclidean distance in joint space.

- One strategy to improve path quality is to post-process paths via "shortcutting", which tries to replace existing portions of a path with shorter segments
Geraerts04 . This is often implemented with the following algorithm: 1) Randomly select two non-consecutive nodes along the path. 2) Try to connect them with a RRT's extend operator. 3) If the resulting path is better, replace the existing path with the new, better path. Steps 1-3 are repeated until a termination condition (often a finite number of steps or time). For this problem, we can assume that the extend operator is a straight line in joint space. Consider the graph below, where RRT has found a rather jerky path from $q_{start}$ to $q_{goal}$. There is an obstacle (shown in red) and $q_{start}$ and $q_{goal}$ are highlighted in blue (disclaimer: This graph was manually generated to serve as an illustrative example). Name one pair of nodes for which the shortcutting algorithm would result in a shorter path (i.e. two nodes along our current solution path for which we could produce a shorter path if we were to directly connect them). You should assume the distance metric is the 2D Euclidean distance. - Shortcutting as a post-processing technique, reasons over the existing path and enables local "re-wiring" of the graph. It is a heuristic and does not, however, guarantee that the tree will encode the shortest path. To explore this, let's zoom in one one iteration of RRT (as illustrated below), where $q_{sample}$ is the randomly generated configuration, $q_{near}$ was the closest node on the existing tree and $q_{new}$ is the RRT extension step from $q_{near}$ in the direction of q_sample. When the standard RRT algorithm (which you implemented in a previous exercise) adds $q_{new}$ to the tree, what node is its parent? If we wanted our tree to encode the shortest path from the starting node, $q_{start}$, to each node in the tree, what node should be the parent node of $q_{new}$?

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- "Automated Construction of Robotic Manipulation Programs", PhD thesis, Carnegie Mellon University, August, 2010. ,
- "{CHOMP}: Gradient Optimization Techniques for Efficient Motion Planning", IEEE International Conference on Robotics and Automation (ICRA) , May, 2009. ,
- "{STOMP}: Stochastic trajectory optimization for motion planning", 2011 IEEE international conference on robotics and automation , pp. 4569--4574, 2011. ,
- "A tutorial on Newton methods for constrained trajectory optimization and relations to SLAM, Gaussian Process smoothing, optimal control, and probabilistic inference", Geometric and numerical foundations of movements, pp. 361--392, 2017. ,
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- "Sampling-based Algorithms for Optimal Motion Planning", Int. Journal of Robotics Research, vol. 30, pp. 846--894, June, 2011. ,

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