# Robotic Manipulation

Perception, Planning, and Control

Russ Tedrake

© Russ Tedrake, 2020-2022
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Note: These are working notes used for a course being taught at MIT. They will be updated throughout the Fall 2022 semester.

# Bin Picking

Consider moving / teaching this BEFORE geometric pose estimation. It could actually be nice to introduce cameras / point clouds with this material instead of with pose estimation.

Our initial study of geometric perception gave us some powerful tools, but also revealed some major limitations. In the next chapter, we will begin applying techniques from deep learning to perception. Spoiler alert: those methods are going to be insatiable in their hunger for data. So before we get there, I'd like to take a brief excursion into a nice subproblem that might help us feed that need.

In this chapter we'll consider the simplest version of the bin picking problem: the robot has a bin full of random objects and simply needs to move those objects from one bin to the other. We'll be agnostic about what those objects are and about where they end up in the other bin, but we would like our solution to achieve a reasonable level of performance for a very wide variety of objects. This turns out to be a pretty convenient way to create a training ground for robot learning -- we can set the robot up to move objects back and forth between bins all day and night, and intermittently add and remove objects from the bin to improve diversity. Of course, it is even easier in simulation!

Bin picking has potentially very important applications in industries such as logistics, and there are significantly more refined versions of this problem. For example, we might need to pick only objects from a specific class, and/or place the objects in known position (e.g. for "packing"). But let's start with the basic case.

# Generating random cluttered scenes

If our goal is to test a diversity of bin picking situations, then the first task is to figure out how to generate diverse simulations. How should we populate the bin full of objects? So far we've set up each simulation by carefully setting the initial positions (in the Context) for each of the objects, but that approach won't scale.

So far we've used robot description files (e.g., URDF, SDF, MJCF, or Model Directives) to parse into the MultibodyPlant; those were sufficient for setting up the bins, cameras, and the robot which can be welded in fixed positions. But generating distributions over objects and object initial conditions is more advanced, and we need an algorithm for that.

# Falling things

In the real world, we would probably just dump the random objects into the bin. That's a decent strategy for simulation, too. We can roughly expect our simulation to faithfully implement multibody physics as long as our initial conditions (and time step) are reasonable; the physics isn't well defined if we initialize the Context with multiple objects occupying the same physical space. The simplest and most common way to avoid this is to generate a random number of objects in random poses, with their vertical positions staggered so that they trivially start out of penetration.

If you look for them, you can find animations of large numbers of falling objects in the demo reels for most advanced multibody simulators. (These demos are actually a bit of a gimmick; the simulations may look realistic to us, but they are often not very physically accurate.) For our purposes the falling dynamics themselves are not the focus. We just want the state of the objects where they are done falling as initial conditions for our manipulation system.

maybe cite the falling things paper, but make it clear that the idea is not new?

# Piles of foam bricks in 2D

Here is the 2D case. I've added many instances of our favorite red foam brick to the plant. Note that it's possible to write highly optimized 2D simulators; that's not what I've done here. Rather, I've added a planar joint connecting each brick to the world, and run our full 3D simulator. The planar joint has three degrees of freedom. I've oriented them here to be $x$, $z$, and $\theta$ to constrain the objects to the $xz$ plane.

I've set the initial positions for each object in the Context to be uniformly distributed over the horizontal position, uniformly rotated, and staggered every 0.1m in their initial vertical position. We only have to simulate for a little more than a second for the bricks to come to rest and give us our intended "initial conditions".

It's not really any different to do this with any random objects -- here is what it looks like when I run the same code, but replace the brick with a random draw from a few objects from the YCB dataset Calli17. It somehow amuses me that we can see the central limit theorem hard at work, even when our objects are slightly ridiculous.

# Filling bins with clutter

The same technique also works in 3D. Setting uniformly random orientations in 3D requires a little more thought, but Drake supplies the method UniformlyRandomRotationMatrix (and also one for quaternions and roll-pitch-yaw) to do that work for us.

Please appreciate that bins are a particularly simple case for generating random scenarios. If you wanted to generate random kitchen environments, for example, then you won't be as happy with a solution that drops refrigerators, toasters, and stools from uniformly random i.i.d. poses. In those cases, authoring reasonable distributions gets much more interesting; we will revisit the topic of generative scene models Izatt20 later in the notes.

# Static equilibrium with frictional contact

Even in the case of bins, we should try to think critically about whether dropping objects from a height is really the best solution. Given our discussion in the last chapter about writing optimizations with non-penetration constraints, I hope you are already asking yourself: why not use those constraints again here? Let's explore that idea a bit further.

I won't dive into a full discussion of multibody dynamics nor multibody simulation, though I do have more notes available here. What is important to understand here is that the equations of motion of our multibody system are described by differential equations of the form: $$M(q)\dot{v} + C(q,v)v = \tau_g(q) + \sum_i J^T_i(q)f^{c_i}.$$ We already understand the generalized positions, $q$, and velocities, $v$. The left side of the equation is just a generalization of "mass times acceleration", with the mass matrix, $M$, and the Coriolis terms $C$. The right hand side is the sum of the (generalized) forces, with $\tau_g(q)$ capturing the terms due to gravity, and $f^{c_i}$ is the spatial force due to the $i$th contact. $J_i(q)$ is the $i$th "contact Jacobian" -- it is the Jacobian that maps from the generalized velocities to the spatial velocity of the $i$th contact frame.

Our interest here is in finding (stable) steady-state solutions to these differential equations that can serve as good initial conditions for our simulation. At steady-state we have $v=\dot{v}=0$, and conveniently all of the terms on the left-hand side of the equations are zero. This leaves us with just the force-balance equations $$\tau_g(q) = - \sum_i J^T_i(q) f^{c_i}.$$ But we still need to understand where the contact forces come from.

# Collision geometry

Geometry engines for robotics, like SceneGraph in , distinguish between a few different roles that geometry can play in a simulation. In robot description files, we distinguish between visual and collision geometries. In particular, every rigid body in the simulation can have multiple collision geometries associated with it (playing the "proximity" role). Collision geometries are often much simpler than the visual geometries we use for illustration and simulating perception -- sometimes they are just a low-polygon-count version of the visual mesh and sometimes we actually use much simpler geometry (like boxes, spheres, and cylinders). These simpler geometries make the physics engine faster and more robust.

For subtle reasons we will explore below, in addition to simplifying the geometry, we sometimes over-parameterize the collision geometry in order to make the numerics more robust. For example, when we simulate the red brick, we actually use nine collision geometries for the one body. The (exaggerated) contact geometry used for robust simulation of boxes. We add contact "points" (epsilon radius spheres) to each corner, and have a slightly inset box for the remaining contacts. Here is the interactive version.

# Collision geometry inspector

Drake's tutorial on "authoring a multibody simulation" has a very useful model_inspector tool that publishes by the illustration and collision geometry roles to Meshcat. This is very useful for designing new models, but also for understanding the contact geometry of existing models. You might want to copy that snippet of code into your own notebook to use it with your own model files. (We'll be making this much easier soon).

SceneGraph also implements the concept of a collision filter. It can be important to specify that, for instance, the iiwa geometry at link 5 cannot collide with the geometry in links 4 or 6. Specifying that some collisions should be ignored not only speeds up the computation, but it also facilitates the use of simplified collision geometry for links. It would be extremely hard to approximate link 4 and 5 accurately with spheres, and cylinders if I had to make sure that those spheres and cylinders did not overlap in any feasible joint angle. The default collision filter settings should work fine for most applications, but you can tweak them if you like.

So where do the contact forces, $f^{c_i}$, come from? There is potentially an equal and opposite contact force for every pair of collision geometries that are not filtered out by the collision filter. In SceneGraph, the GetCollisionCandidates method returns them all. We'll take to calling the two bodies in a collision pair "body A" and "body B".

# The Contact Frame

We still need to decide the magnitude of these spatial forces, and to write down the magnitude, we need to specify the contact frame in which the spatial force is to be applied. For instance, we might use $C_B$ to denote a contact frame on body $B$, with the forces applied at the origin of the frame.

Most simulators summarize the contact between two bodies as a translational force (e.g. zero torque) at one or more contact points. Our convention will be to align the positive $z$ axis with the "contact normal", with positive forces resisting penetration. Defining this normal can be deceptively complicated. For instance, what is the normal at the corner of a box? Taking the normal as a (sub-)gradient of the signed-distance function for the collision geometry provides a reliable definition that will extend to the distance between two bodies and into generalized coordinates. The $x$ and $y$ axes of the contact frame are any orthogonal basis for the tangential coordinates. You can find additional figures and explanations here.

# Brick on a half-plane

Let's work through these details on a simple example -- our foam brick sitting on the ground. The ground is welded to the world, so has no degrees of freedom; we can ignore the forces applied to the ground and focus on the forces applied to the brick.

Where are the contact forces and contact frames? If we used only box-on-box geometry, then the locations of the contact forces are ambiguous during this "face-on-face" contact; this is even more true in 3D (where three forces would be sufficient). But by adding extra contact spheres to the corners of our box, we are telling the physics engine that we specifically want contact forces at each of the corners (and all four of the corners in 3D). I've labeled the frames $C_1$ and $C_2$ in the sketch below.

In order to achieve static equilibrium, we require that all of the forces and torques on the brick be in balance. This is a simple manifestation of the equations $$\tau_g(q) = - \sum_i J^T_i(q) f^{c_i},$$ because the configuration $q$ is the $x,z, \theta$ position and orientation of the brick; we have three equations and three unknowns: \begin{gather*} 0 = \sum_i f^{c_i}_{C_{i,x}} \\ -mg = -\sum_i f^{c_i}_{C_{i,z}} \\ 0 = \sum_i \left[ \left[ p^{C_i} - p^{B_{cm}} \right] \times f^{c_i} \right]_{W_y}.\end{gather*} The last equation represents the torque balance, taken around the center of mass of the brick which I've call $p^{B_{cm}}$. Torque about $\theta$ corresponds to the $y$ component of the cross product. The torque balance ensures that $f^{c_1}_{C_{1,z}} = f^{c_2}_{C_{2,z}}$ assuming the center of mass is in the middle of the box; force balance in $z$ set them equal to $\frac{mg}{2}$. We haven't said anything about the horizontal forces yet ($0$ is a reasonable solution here). Let's develop that next.

# The (Coulomb) Friction Cone

Now the rules governing contact forces can begin to take shape. First and foremost, we demand that there is no force at a distance. Using $\phi_i(q)$ to denote the distance between two bodies in configuration $q$, we have $$\phi(q) > 0 \Rightarrow f^{c_i} = 0.$$ Second, we demand that the normal force only resists penetration; bodies are never pulled into contact: $$f^{c_i}_{C_z} \ge 0.$$ In rigid contact models, we solve for the smallest normal force enforces the non-penetration constraint (this is known as the principle of least constraint). In soft contact models, we define the force to be a function of the penetration depth and velocity.

Citations Need figures here

Forces in the tangential directions are due to friction. The most commonly used model of friction is Coulomb friction, which states that $$|f^{c_i}_{C_{x,y}}|_2 \le \mu f^{c_i}_{C_z},$$ with $\mu$ a non-negative scalar coefficient of friction. Typically we define both a $\mu_{static}$, which is applied when the tangential velocity is zero, and $\mu_{dynamic}$, applied when the tangential velocity is non-zero. In the Coulomb friction model, the tangential contact force is the force within this friction cone which produces maximum dissipation.

Taken together, the geometry of these constraints forms a cone of admissable contact forces. It is famously known as the "friction cone", and we will refer to it often.

# Brick on an inclined half-plane

If we take our last example, but tilt the table to an angle relative to gravity, then the horizontal forces start becoming important. Before going through the equations, let's check your intuition. Will the magnitude of the forces on the two corners stay the same in this configuration? Or will there be more contact force on the lower corner?

In the illustration above, you'll notice that the contact frames have rotated so that the $z$ axis is aligned with the contact normals. I've sketched two possible friction cones (dark green and lighter green), corresponding to two different coefficients of friction. We can tell immediately by inspection that the smaller value of $\mu$ (corresponding to the darker green) cannot produce contact forces that will completely counteract gravity (the friction cone does not contain the vertical line). In this case, the box will slide and no static equilibrium exists in this configuration.

If we increase the coefficient of (static) friction to the range corresponding to the lighter green, then we can find contact forces that produce an equilibrium. Indeed, for this problem, we need some amount of friction to even have an equilibrium (we'll explore this in the exercises). We also need for the vertical projection of the center of mass onto the ramp to land between the two contact points, otherwise the brick will tumble over the bottom edge. We can see this by writing our same force/torque balance equations. We can write them in body frame, B, assuming the center of mass in the center of the brick and the brick has length $l$ and height $h$:\begin{gather*} f^{c_1}_{B_x} + f^{c_2}_{B_x} = - mg\sin\gamma \\ f^{c_1}_{B_z} + f^{c_2}_{B_z} = mg\cos\gamma \\ -h f^{c_1}_{B_x} + l f^{c_1}_{B_z} = h f^{c_2}_{B_x} + l f^{c_2}_{B_z} \\ f^{c_1}_{B_z} \ge 0, \quad f^{c_2}_{B_z} \ge 0 \\ |f^{c_1}_{B_x}| \le \mu f^{c_1}_{B_z}, \quad |f^{c_2}_{B_x}| \le \mu f^{c_2}_{B_z} \end{gather*}

So, are the magnitude of the contact forces the same or different? Substituting the first equation into the third reveals $$f^{c_2}_{B_z} = f^{c_1}_{B_z} + \frac{mgh}{l}\sin\gamma.$$

# Static equilibrium as an optimization problem

Rather than dropping objects from a random height, perhaps we can initialize our simulations using optimization to find the initial conditions that are already in static equilibrium. In , the StaticEquilbriumProblem collects all of the constraints we enumerated above into an optimization problem: \begin{align*} \find_q \quad \subjto \quad& \tau_g(q) = - \sum_i J^T_i(q) f^{c_i} & \\ & f^{c_i}_{C_z} \ge 0 & \forall i, \\ & |f^{c_i}_{C_{x,y}}|_2 \le \mu f^{c_i}_{C_z} & \forall i, \\ & \phi_i(q) \ge 0 & \forall i, \\ & \phi(q) = 0 \textbf{ or } f^{c_i}_{C_z} = 0 &\forall i, \\ & \text{joint limits}.\end{align*} This is a nonlinear optimization problem: it includes the nonconvex non-penetration constraints we discussed in the last chapter. The second-to-last constraints (a logical or) is particularly interesting; constraints of the form $x \ge 0, y \ge 0, x =0 \textbf{ or } y = 0$ are known as complementarity constraints, and are often written as $x \ge 0, y \ge 0, xy = 0$. We can make the problem easier for the nonlinear optimization solver by relaxing the equality to $0 \le \phi(q) f^{c_i}_{C_z} \le \text{tol}$, which provides a proper gradient for the optimizer to follow at the cost of allowing some force at a distance.

It's easy to add additional costs and constraints; for initial conditions we might use an objective that keeps $q$ close to an initial configuration-space sample.

# Tall Towers

So how well does it work?

finish this...

# Contact force inspector

I've created a simple GUI that allows you to pick any two primitive geometry types and inspect the contact information that is computed when those object are put into penetration. There is a lot of information displayed there! Take a minute to make sure you understand the colors, and the information provided in the textbox display.

When I play with the GUI above, I feel almost overwhelmed. First, overwhelmed by the sheer number of cases that we have to get right in the code; it's unfortunately extremely common for open-source tools to have bugs in here. But second, overwhelmed by the sense that we are asking the wrong question. Asking to summarize the forces between two bodies in deep penetration with a single Cartesian force applied at a point is fraught with peril. As you move the objects, you will find many discontinuities; this is a common reason why you sometimes see rigid body simulations "explode". It might seem natural to try to use multiple contact points instead of a single contact point to summarize the forces, and some simulators do, but it is very hard to write an algorithm that only depends on the current configurations of the geometry which applies forces consistently from one time step to the next with these approaches.

I currently feel that the only fundamental way to get around these numerical issues is to start reasoning about the entire volume of penetration. The Drake developers have recently proposed a version of this that we believe we can make computationally tractable enough to be viable for real-time simulation Elandt19, and are working hard to bring this into full service in Drake. You will find many references to "hydroelastic" contact throughout the code. We hope to turn it on by default soon.

Add the hydroelastic skittles video here. https://youtu.be/5-k2Pc6OmzE

In the mean time, we can make simulations robust by carefully curating the contact geometry...

# Grasp selection

Now that we have that ability to populate our bins full of random clutter, we need to determine where we will grasp. The pipeline that one might expect, and indeed the pipeline that robotics was somewhat stuck on for decades, is that the first step should be to write some form of perception system to look into the bin and segment the point cloud into individual objects. These days that's not impossible, but it's still hard!

The newer approach that has proven incredibly useful for bin picking is to just look for graspable areas directly on the (unsegmented) point cloud. Some of the most visible proponents of this approach are tenPas17+Mahler17+Zeng18. If you look at those references, you'll see that they all use learning to somehow decide where in the point cloud to grasp. And I do think learning has a lot to offer in terms of choosing good grasps -- it's a nice and natural learning formulation and there is significant information besides just what appears in the immediate sensor returns that can contribute to deciding where to grasp. But often you will see that underlying those learning approaches is an approach to selecting grasps based on geometry alone, if only to produce the original training data. I also think that the community doesn't regularly acknowledge just how well the purely geometric approaches still work. I'd like to discuss those first.

# Point cloud pre-processing

To get a good view into the bin, we're going to set up multiple RGB-D cameras. I've used three per bin in all of my examples here. And those cameras don't only see the objects in the bin; they also see the bins, the other cameras, and anything else in the viewable workspace. So we have a little work to do to merge the point clouds from multiple cameras into a single point cloud that only includes the area of interest.

First, we can crop the point cloud to discard any points that are from outside the area of interest (which we'll define as an axis-aligned bounding box immediately above the known location of the bin).

As we will discuss in some detail below, many of our grasp selection strategies will benefit from estimating the "normals" of the point cloud (a unit vector that estimates the normal direction relative to the surface of the underlying geometry). It is actually better to estimate the normals on the individual point clouds, making use of the camera location that generated those points, than to estimate the normal after the point cloud has been merged.

For sensors mounted on the real world, merging point clouds requires high-quality camera calibration and must deal with the messy depth returns. All of the tools from the last chapter are relevant, as the tasks of merging the point clouds is another instance of the point-cloud-registration problem. For the perfect depth measurements we can get out of simulation, given known camera locations, we can skip this step and simply concatenate the list of points in the point clouds together.

Finally, the resulting raw point clouds likely include many more points then we actually need for our grasp planning. One of the standard approaches for down-sampling the point clouds is using a voxel grid -- regularly sized cubes tiled in 3D. We then summarize the original point cloud with exactly one point per voxel (see, for instance Open3D's note on voxelization). Since point clouds typically only occupy a small percentage of the voxels in 3D space, we use sparse data structures to store the voxel grid. In noisy point clouds, this voxelization step is also a useful form of filtering.

# Mustard bottle point clouds

I've produced a scene with three cameras looking at our favorite YCB mustard bottle. I've taken the individual point clouds (already converted to the world frame by the DepthImageToPointCloud system), cropped the point clouds (to get rid of the geometry from the other cameras), estimated their normals, merged the point clouds, then down-sampled the point clouds. The order is important!

I've pushed all of the point clouds to meshcat, but with many of them set to not be visible by default. Use the drop-down menu to turn them on and off, and make sure you understand basically what is happening on each of the steps. For this one, I can also give you the meshcat output directly, if you don't want to run the code.

# Estimating normals and local curvature

The grasp selection strategy that we will develop below will be based on the local geometry (normal direction and curvature) of the scene. Understanding how to estimate those quantities from point clouds is an excellent exercise in point cloud processing, and is representative of other similar point cloud algorithms.

Let's think about the problem of fitting a plane, in a least-squares sense, to a set of pointsShakarji98. We can describe a plane in 3D with a position $p$ and a unit length normal vector, $n$. The distance between a point $p^i$ and a plane is simply given by the magnitude of the inner product, $\left| (p^i - p)^T n \right|.$ So our least-squares optimization becomes $$\min_{p, n} \quad \sum_{i=1}^N \left| (p^i - p)^T n \right|^2, \quad \subjto \quad |n| = 1.$$ Taking the gradient of the Lagrangian with respect to $p$ and setting it equal to zero gives us that $$p^* = \frac{1}{N} \sum_{i=1}^N p^i.$$ Inserting this back into the objective, we can write the problem as $$\min_n n^T W n, \quad \subjto \quad |n|=1, \quad \text{where } W = \left[ \sum_i (p^i - p^*) (p^i - p^*)^T \right].$$ Geometrically, this objective is a quadratic bowl centered at the origin, with a unit circle constraint. So the optimal solution is given by the (unit-length) eigenvector corresponding to the smallest eigenvalue of the data matrix, $W$. And for any optimal $n$, the "flipped" normal $-n$ is also optimal. We can pick arbitrarily for now, and then flip the normals in a post-processing step (to make sure that the normals all point towards the camera origin).

What is really interesting is that the second and third eigenvalues/eigenvectors also tell us something about the local geometry. Because $W$ is symmetric, it has orthogonal eigenvectors, and these eigenvectors form a (local) basis for the point cloud. The smallest eigenvalue pointed along the normal, and the largest eigenvalue corresponds to the direction of least curvature (the squared dot product with this vector is the largest). This information can be very useful for finding and planning grasps. tenPas17 and others before them use this as a primary heuristic in generating candidate grasps.

In order to approximate the local curvature of a mesh represented by a point cloud, we can use our fast nearest neighbor queries to find a handful of local points, and use this plane fitting algorithm on just those points. When doing normal estimation directly on a depth image, people often forgo the nearest-neighbor query entirely; simply using the approximation that neighboring points in pixel coordinates are often nearest neighbors in the point cloud. We can repeat that entire procedure for every point in the point cloud.

I remember when working with point clouds started to become a bigger part of my life, I thought that surely doing anything moderately computational like this on every point in some dense point cloud would be incompatible with online perception. But I was wrong! Even years ago, operations like this one were often used inside real-time perception loops. (And they pale in comparison to the number of FLOPs we spend these days evaluating large neural networks).

# Normals and local curvature of the mustard bottle.

I've coded up the basic least-squares surface estimation algorithm, with the query point in green, the nearest neighbors in blue, and the local least squares estimation drawn with our RGB$\Leftrightarrow$XYZ frame graphic. You should definitely slide it around and see if you can understand how the axes line up with the normal and local curvature.

You might wonder where you can read more about algorithms of this type. I don't have a great reference for you. But Radu Rusu was the main author of the point cloud libraryRusu11, and his thesis has a lot of nice summaries of the point cloud algorithms of 2010Rusu10.

# Evaluating a candidate grasp

Now that we have processed our point cloud, we have everything we need to start planning grasps. I'm going to break that discussion down into two steps. In this section we'll come up with a cost function that scores grasp candidates. In the following section, we'll discuss some very simple ideas for trying to find grasps candidates that have a low cost.

There is a massive literature in robotics on "grasp metrics" and "grasp analysis". The Handbook of RoboticsSiciliano16 has an entire chapter on grasping devoted to it (which I recommend) Prattichizzo08. For a course on manipulation (especially from someone who cares a lot about dynamics) to spend so little time on this topic is perhaps surprising, but a great deal of that literature is focused on analyzing point contacts between a dexterous hand and an object with known geometry. That's simply not the situation we find ourselves in for a lot of manipulation work today. We have ourselves a simple two fingered grippers with relatively a lot of grasping force (> 80 N) and only local point clouds to represent our geometry.

But there is one key idea that I do want to extract from that literature: the value of antipodal grasps. Roughly speaking, once we pick up an object -- or whatever happens to be between our fingers when we squeeze -- then we will expect the contact forces between our fingers to have to resist at least the gravitational wrench (the spatial force due to gravity) of the object. The closing force provided by our gripper is in the gripper's $x$-axis, but if we want to be able to pick up the object without it slipping from our hands, then we need forces inside the friction cones of our contacts to be able to resist the gravitational wrench. Since we don't know what that wrench will be (and are somewhat constrained by the geometry of our fingers), a reasonable strategy is to look for places on the point cloud where the normals will align with $x$-axis of the gripper. That means finding a pair of points on the point cloud with normals that are collinear with the line connecting the points and are pointing in opposite directions, these are antipodal point pairs. In a real point cloud, we are unlikely to find perfect antipodal pairs, but finding areas with normals pointing in nearly opposite directions is a good strategy for grasping!

# Scoring grasp candidates

In practice, the contact between our fingers and the object(s) will be better described by a patch contact than by a point contact (due to the deflection of the rubber fingertips and any deflection of the objects being picked). So it makes sense to look for patches of points with agreeable normals. There are many ways one could write this, I've done it here by transforming the processed point cloud of the scene into the candidate frame of the gripper, and cropped away all of the points except the ones that are inside a bounding box between the finger tips (I've marked them in red in MeshCat). The first term in my grasping cost function is just reward for all of the points in the point cloud, based on how aligned their normal is to the $x$-axis of the gripper: $$\text{cost} = -\sum_i (n^i_{G_x})^2,$$ where $n^i_{G_x}$ is the $x$ component of the $i$th point in the cropped point cloud expressed in the gripper frame.

There are other considerations for what might make a good grasp, too. For our kinematically limited robot reaching into a bin, we might favor grasps that put the hand in favorable orientation for the arm. In the grasp metric I've implemented in the code, I added a cost for the hand deviating from vertical. I can reward the dot product of the vector world $-z$ vector, $[0, 0, -1]$ with the $y$-axis in gripper frame rotated into world frame with : $$\text{cost} \mathrel{{+}{=}} -\alpha \begin{bmatrix} 0 & 0 &-1\end{bmatrix}R^G \begin{bmatrix}0 \\ 1 \\ 0\end{bmatrix} = \alpha R^G_{3,2},$$ where $\alpha$ is relative cost weight, and $R^G_{3,2}$ is the scalar element of the rotation matrix in the third row and second column.

Finally, we need to consider collisions between the candidate grasp and both the bins and with the point cloud. I simply return infinite cost when the gripper is in collision. I've implemented all of those terms in the notebook, and given you a sliders to move the hand around and see how the cost changes.

# Generating grasp candidates

We've defined a cost function that, given a point cloud from the scene and a model of the environment (e.g. the location of the bins), can score a candidate grasp pose, $X^G$. So now we would like to solve the optimization problem: find $X^G$ that minimizes the cost subject to the collision constraints.

Unfortunately, this is an extremely difficult optimization problem, with a highly nonconvex objective and constraints. Moreover, the cost terms corresponding to the antipodal points is zero for most $X^G$ -- since most random $X^G$ will not have any points between the fingers. As a result, instead of using the typical mathematical programming solvers, most approaches in the literature resort to a randomized sampling-based algorithm. And we do have strong heuristics for picking reasonable samples.

One heuristic, used for instance in tenPas17, is to use the local curvature of the point cloud to propose grasp candidates that have the point cloud normals pointing into the palm, and orients the hand so that the fingers are aligned with the direction of maximum curvature. One can move the hand in the direction of the normal until the fingers are out of collision, and even sample nearby points. We have written an exercise for you to explore this heuristic. But for our YCB objects, I'm not sure it's the best heuristic; we have a lot of boxes, and boxes don't have a lot of information to give in their local curvature.

Another heuristic is to find antipodal point pairs in the point cloud, and then sample grasp candidates that would align the fingers with those antipodal pairs. Many 3D geometry libraries support "ray casting" operations at least for a voxel representation of a point cloud; so a reasonable approach to finding antipodal pairs is to simply choose a point at random in the point cloud, then ray cast into the point cloud in the opposite direction of the normal. If the normal of the voxel found by ray casting is sufficiently antipodal, and if the distance to that voxel is smaller than the gripper width, then we've found a reasonable antipodal point pair.

# Generating grasp candidates

As an alternative to ray casting, I've implemented an even simpler heuristic in my code example: I simply choose a point at random, and start sampling grasps in orientation (starting from vertical) that align the $x$-axis of the gripper with the normal of that point. Then I mostly just rely on the antipodal term in my scoring function to allow me to find good grasps.

I do implement one more heuristic -- once I've found the points in the point cloud that are between the finger tips, then I move the hand out along the gripper $x$-axis so that those points are centered in the gripper frame. This helps prevent us knocking over objects as we close our hands to grasp them.

But that's it! It's a very simple strategy. I sample a handful of candidate grasps and just draw the top few with the lowest cost. If you run it a bunch of times, I think you will find it's actually quite reasonable. Every time it runs, it is simulating the objects falling from the sky; the actual grasp evaluation is actually quite fast.

# The corner cases

If you play around with the grasp scoring I've implemented above a little bit, you will find deficiencies. Some of them are addressed easily (albeit heuristically) by adding a few more terms to the cost. For instance, I didn't check collisions of the pre-grasp configuration, but this could be added easily.

There are other cases where grasping alone is not sufficient as a strategy. Imagine that you place an object right in one of the corners of the bin. It might not be possible to get the hand around both sides of the object without being in collision with either the object or the side. The strategy above will never choose to try to grab the very corner of a box (because it always tried to align the sample point normal with the gripper $x$), and it's not clear that it should. This is probably especially true for our relatively large gripper. In the setup we used at TRI, we implemented an additional simple "pushing" heuristic that would be used if there were point clouds in the sink, but no viable grasp candidate could be found. Instead of grasping, we would drive the hand down and nudge that part of the point cloud towards the middle of the bin. This can actually help a lot!

There are other deficiencies to our simple approach that would be very hard to address with a purely geometric approach. Most of them come down to the fact that our system so far has no notion of "objects". For instance, it's not uncommon to see this strategy result in "double picks" if two objects are close together in the bin. For heavy objects, it might be important to pick up the object close to the center of mass, to improve our chances of resisting the gravitational wrench while staying in our friction cones. But our strategy here might pick up a heavy hammer by squeezing just the very far end of the handle.

Interestingly, I don't think the problem is necessarily that the point cloud information is insufficient, even without the color information. I could show you similar point clouds and you wouldn't make the same mistake. These are the types of examples where learning a grasp metric could potentially help. We don't need to achieve artificial general intelligence to solve this one; just experience knowing that when we tried to grasp in someplace before we failed would be enough to improve our grasp heuristics significantly.

render some point clouds that a human can distinguish but out algorithm would not.

# A simple state machine

Maybe add a section here about monte-carlo testing?

# Assessing static equilibrium

For this problem, we'll use the Coulomb friction model, where $|f_t| \le \mu f_n$. In other words, the friction force is large enough to resist any movement up until the force required would exceed $\mu f_n$, in which case $|f_t| = \mu f_n$.

1. Consider a box with mass $m$ sitting on a ramp at angle $\theta$, with coefficient of box $\mu$ in between the sphere and the ramp:

For a given angle $\theta$, what is the minimum coefficient of friction required for the box to not slip down the plane? Use $g$ for acceleration due to gravity.

Now consider a flat ground plane with three solid (uniform density) spheres sitting on it, with radius $r$ and mass $m$. Assume they have the same coefficient of friction $\mu$ between each other as with the ground.

For each of the following configurations: could the spheres be in static equilibrium for some $\mu\in[0,1]$, $m > 0$, $r > 0$? Explain why or why not. Remember that both torques and forces need to be balanced for all bodies to be in equilibrium.

To help you solve these problems, we have to help you build intuition and test your answers. It lets you specify the configuration of the spheres and then uses the StaticEquilbriumProblem class to solve for static equilibrium. Use this notebook to help visualize and think about the system, but for each of the configurations, you should have a physical explanation for your answer. (An example of such a physical explanation would be a free body diagram of the forces and torques on each sphere, and equations for how they can or cannot sum to zero. This is essentially what StaticEquilbriumProblem checks for.)

1. Spheres stacked on top of each other:
2. One sphere on top of another, offset:
3. Spheres stacked in a pyramid:
4. Spheres stacked in a pyramid, but with a distance $d$ in between the bottom two:

Finally, a few conceptual questions on the StaticEquilbriumProblem:

1. Why does it matter what initial guess we specify for the system? (Hint: what type of optimization problem is this?)
2. Take a look at the Drake documentation for StaticEquilbriumProblem. It lists the constraints that are used when it's solve for equilibrium. Which two of these can a free body diagram answer?

# Normal Estimation from Depth

For this exercise, you will investigate a slightly different approach to normal vector estimation. In particular, we can exploit the spatial structure that is already in a depth image to avoid computing nearest neighbors. You will work exclusively in . You will be asked to complete the following steps:

1. Implement a method to estimate normal vectors from a depth image, without computing nearest neighbors.
2. Reason about a scenario where the depth image-based solution will not be as performant as computing nearest-neighbors.

# Analytic Antipodal Grasping

So far, we have used sampling-based methods to find antipodal grasps - but can we have find one analytically if we knew the equation of the object shape? For this exercise, you will analyze and implement a correct-by-construction method to find antipodal points using symbolic differentiation and MathematicalProgram. You will work exclusively in . You will be asked to complete the following steps:

1. Prove that antipodal points are critical points of an energy function defined on the shape.
2. Prove the converse does not hold.
3. Implement a method to find these antipodal points using MathematicalProgram.
4. Analyze the Hessian of the energy function and its relation to the type of antipodal grasps.

# Grasp Candidate Generation

In the chapter Colab notebook, we generated grasp candidates using the antipodal heuristic. In this exercise, we will investigate an alternative method for generating grasp candidates based on local curvature, similar to the one proposed in tenPas17. This exercise is implementation-heavy, and you will work exclusively in .

# Behavior Trees

Let's reintroduce the terminology of behavior trees. Behavior trees provide a structure for switching between different tasks in a way that is both modular and reactive. Behavior trees are a directed rooted tree where the internal nodes manage the control flow and the leaf nodes are actions to be executed or conditions to be evaluated. For example, an action may be to "pick ball". This action can succeed or fail. A condition may be "Is my hand empty?" which can be true (thus the condition succeeds) or can be false (the condition fails).

We'll consider two categories of control flow nodes:

Sequence Node: ($\rightarrow$)
Sequence nodes execute each of the child behaviors one after another. The sequence fails if any of the children fail. One way to think about this operator is that a sequence node takes an "and" over all of the child behaviors.
Fallback Node: (?)
Fallback nodes also execute each of the child behaviors one after another. However, fallback succeeds if any of the children succeed. One way to think about this operator is that the fallback node takes an "or" over all of the children behaviors.
The symbols are visualized below. Sequence nodes are represented as an arrow in a box, fallback nodes are represented as a question mark in a box, actions are inside boxes and conditions are inside ovals.

Let's apply our understanding of behavior trees in the context of a simple task where a robot's goal is to: find a ball, pick the ball up and place the ball in a bin. We can describe task with the high-level behavior tree:

Confirm to yourself that this small behavior tree captures the task! We can go one level deeper and expand out our "Pick Ball" behavior:

The pick ball behavior can be considered as such: Let's start with the left branch. If the ball is already close, the condition "Ball Close?" returns true. If the ball is not close, we execute an action to "Approach ball", thus making the ball close. Thus either the ball is already close (i.e. the condition "Ball Close?" is true) or we approach the ball (i.e. the action "Approach ball" is successfully executed) to make the ball close. Given that the ball is now close, we consider the right branch of the pick action. If the ball is already grasped, the condition "Ball Grasped?" returns true. if the ball is not grasped, we execute the action, "Grasp Ball".
We can expand our our behavior tree even further to give our full behavior:

Here we've added the behavior that if the robot cannot complete the task, it can ask a kind human for help. Take a few minutes to walk through the behavior tree and convince yourself it encodes the desired task behavior.

1. We claimed behavior trees enable reactive behavior. Let's explore that. Imagine we have a robot executing our behavior tree. As the robot is executing the action "Approach Bin", a rude human knocks the ball out of the robot's hand. The ball rolls to a position that the robot can still see, but is quite far away. Following the logic of the behavior tree, what condition fails and what action is executed because of this?
2. Another way to mathematically model computation is through finite state machines (FSM). Finite state machines are composed of states, transitions and events. We can encode the behavior of our behavior tree for our pick-up task through a finite state machine, as shown below: We have left a few states and transitions blank. Your task is to fill in those 4 values such that the finite state machine produces the same behavior as the behavior tree.

From this simple task we can see that while finite state machines can often be easy to understand and intuitive, they can quickly get quite messy!

1. Throughout this chapter and the lectures we've discussed a bin picking task. Our goal is to now capture the behavior of the bin-picking task using a behavior tree. We want to encode the following behavior: while Bin 1 is not empty, we are going to select a feasible grasp within the bin, execute our grasp and transport what's within our grasp to Bin 2. If there are still objects within Bin 1, but we cannot find a feasible grasp, assume we have some action that shakes the bin, hence randomizing the objects and (hopefully) making feasible grasps available.

We define the following conditions:
• "Bin 1 empty?
• "Feasible Grasp Exists?"
• "Grasp selected?"
• "Grasped Object(s) above Bin 2?"
• "Object Grasped?"
• "Object(s) in Bin 2?"

And the following actions:
• "Drop grasped object(s) into Bin 2"
• "Grasp Object(s)"
• "Select Grasp"
• "Shake Bin"
• "Transport grasped object(s) to above Bin 2"

Using these conditions, actions and our two control flow nodes (Sequence and Fallback), draw a behavior tree that captures the behavior of our bin picking task.

We claimed behavior trees are modular. Their modularity comes from the fact that we can write reusable snippets of code that define actions and conditions and then compose them using behavior trees. For example, we can imagine that the "Select Grasp" action from part (c) is implemented using the antipodal grasping mechanism we have built up throughout this chapter. And the "Transport grasped object(s) to above Bin 2" action could be implemented using the pick and place tools we developed in Chapter 3.

Source: The scenario and figures for parts (a) and (b) of this exercise are inspired by material in Colledanchise18