# 4. Computational Motor Control: Kinematics

## Table of Contents

## Introduction

Here we will talk about kinematic models of the arm. We will start
with single-joint (elbow) models and progress to two-joint (shoulder,
elbow) and even three-joint (shoulder, elbow, wrist) arm models. First
we will talk about kinematic coordinate transformations. The goal is
to develop the equations for forward and inverse kinematic
transformations. **Forward kinematic equations** take us from intrinsic
to extrinsic variables, for example from joint angles to hand
position. So given a 2-element vector of shoulder and elbow angles
\((\theta_{1},\theta_{2})\), we can compute what the 2D location of the
hand is, in a cartesian coordinate system \((H_{x},H_{y})\). **Inverse
kinematic equations** take us in the other direction, from extrinsic
variables to intrinsic variables, e.g. from hand coordinates to joint
angles.

## One-joint Elbow Kinematic Model

Let's first consider a simple model of an arm movement. To begin with we will model a single joint, the elbow joint. Our model will have a Humerus bone (upper arm) and an Ulna bone (lower arm). Our model will be a planar 2D model and so we will ignore pronation and supination of the forearm, and therefore we'll ignore the Radius bone. The lower arm will include the hand, and for now we will assume that the wrist joint is fixed, i.e. it cannot move. For now let's even ignore muscles.

Figure 1: Schematic of a simple kinematic model of the elbow joint

The intrinsic variable in our one-joint elbow model is simple the elbow angle \(\theta\), and the extrinsic variable is the hand position in cartesian coordinates, which is defined by two values, \((H_{x},H_{y})\).

### Forward Kinematics

What are the forward kinematic equations for our elbow model? We simply need to recall our trigonometry from high school (remember SOH-CAH-TOA?):

\begin{eqnarray} H_{x} &= &l \cos(\theta) \\ H_{y} &= &l \sin(\theta) \end{eqnarray}So if I told you that my lower arm (Ulna bone) is 0.45 metres long and that my elbow angle is 35 degrees (remember to convert to radians before using \(\sin()\) and \(\cos()\), and asked you where my hand is located, you could compute this:

\begin{eqnarray} H_{x} &= &l \cos(\theta) &= 0.45 \cos\left(\frac{35*\pi}{180}\right) = 0.37 m\\ H_{y} &= &l \sin(\theta) &= 0.45 \sin\left(\frac{35*\pi}{180}\right) = 0.26 m\\ \end{eqnarray}So the hand position would be \((0.37,0.26)\) (measured in metres).

### Inverse Kinematics

What about the inverse kinematics equations? If I give you the position of my hand, and the length of my lower arm, how can you compute my elbow joint angle? Remember SOH-CAH-TOA: \(\tan()\) equals opposite (O) over adjacent (A) which in our case is \(H_{y}\) over \(H_{x}\) so:

\begin{equation} \tan(\theta) = \frac{H_{y}}{H_{x}} \end{equation}and so:

\begin{equation} \theta = \arctan \left( \frac{H_{y}}{H_{x}} \right) \end{equation}Remember, angles are measured in radians. There are \(2\pi\) radians in 360 degrees.

We can of course do more than compute single values, we can compute entire trajectories. So for example if I gave you a recording of elbow angles over time, for example sampled at 200 Hz (200 samples per second) then using the forward kinematic equations you would be able to plot a cartesian-space (e.g. top-down) trajectory of the hand path, and vice-versa.

# make up an elbow trajectory (pretend this was recorded in an experiment) t = linspace(0,1,200) theta = sin(2*pi*t/4) figure() subplot(2,3,(1,2)) plot(t,theta*180/pi) xlabel('TIME (sec)') ylabel('ELBOW ANGLE (deg)') # compute hand position Hx,Hy l = 0.45 Hx = l * cos(theta) Hy = l * sin(theta) subplot(2,3,(4,5)) plot(t,Hx,'b-') plot(t,Hy,'r-') xlabel('TIME (sec)') ylabel('HAND POSITION (m)') legend(('Hx','Hy'),loc='lower right') subplot(2,3,(3,6)) plot((0,Hx[0]),(0,Hy[0]),'g-') plot((0,Hx[-1]),(0,Hy[-1]),'r-') plot(Hx[0:-1:10],Hy[0:-1:10],'k.') xlabel('X (m)') ylabel('Y (m)') axis('equal')

Figure 2: Example elbow movement

## Two-joint arm

Let's introduce a shoulder joint as well:

Figure 3: Schematic of a simple kinematic model of a two-joint arm

### Forward Kinematics

Now let's think about the forward and inverse kinematic equations to go from intrinsic coordinates, joint angles \((\theta_{1},\theta_{2})\) to extrinsic coordinates \((H_{x},H_{y})\). We will compute the elbow joint position as an intermediate step.

\begin{eqnarray} E_{x} &= &l_{1} \cos(\theta_{1})\\ E_{y} &= &l_{1} \sin(\theta_{1})\\ H_{x} &= &E_{x} + l_{2}\cos(\theta_{1}+\theta_{2})\\ H_{y} &= &E_{y} + l_{2}\sin(\theta_{1}+\theta_{2})\\ \end{eqnarray}Now if I give you a set of shoulder and elbow joint angles \((\theta_{1},\theta_{2})\), you can compute what my hand position \((H_{x},H_{y})\) is.

We can visualize this mapping by doing something like the following: decide on a range of shoulder angles and elbow angles that are physiologically realistic, and sample that range equally in joint space … then run those joint angles through the forward kinematics equations to visualize how those equally-spaced joint angles correspond to cartesian hand positions.

# Function to transform joint angles (a1,a2) to hand position (Hx,Hy) def joints_to_hand(a1,a2,l1,l2): Ex = l1 * cos(a1) Ey = l1 * sin(a1) Hx = Ex + (l2 * cos(a1+a2)) Hy = Ey + (l2 * sin(a1+a2)) return Ex,Ey,Hx,Hy # limb geometry l1 = 0.34 # metres l2 = 0.46 # metres # decide on a range of joint angles n1steps = 10 n2steps = 10 a1range = linspace(0*pi/180, 120*pi/180, n1steps) # shoulder a2range = linspace(0*pi/180, 120*pi/180, n2steps) # elbow # sample all combinations and plot joint and hand coordinates f=figure(figsize=(8,12)) for i in range(n1steps): for j in range(n2steps): subplot(2,1,1) plot(a1range[i]*180/pi,a2range[j]*180/pi,'r+') ex,ey,hx,hy = joints_to_hand(a1range[i], a2range[j], l1, l2) subplot(2,1,2) plot(hx, hy, 'r+') subplot(2,1,1) xlabel('Shoulder Angle (deg)') ylabel('Elbow Angle (deg)') title('Joint Space') subplot(2,1,2) xlabel('Hand Position X (m)') ylabel('Hand Position Y (m)') title('Hand Space') a1 = a1range[n1steps/2] a2 = a2range[n2steps/2] ex,ey,hx,hy = joints_to_hand(a1,a2,l1,l2) subplot(2,1,1) plot(a1*180/pi,a2*180/pi,'bo',markersize=5) axis('equal') xl = get(get(f,'axes')[0],'xlim') yl = get(get(f,'axes')[0],'ylim') plot((xl[0],xl[1]),(a2*180/pi,a2*180/pi),'b-') plot((a1*180/pi,a1*180/pi),(yl[0],yl[1]),'b-') subplot(2,1,2) plot((0,ex,hx),(0,ey,hy),'b-') plot(hx,hy,'bo',markersize=5) axis('equal') xl = get(get(f,'axes')[1],'xlim') yl = get(get(f,'axes')[1],'ylim') plot((xl[0],xl[1]),(hy,hy),'b-') plot((hx,hx),(yl[0],yl[1]),'b-')

Figure 4: Joint vs Hand Kinematics for Two Joint Arm

Note that in the lower plot, the shoulder location is at the origin, \((0,0)\). The blue crosshairs in each subplot correspond to the same arm position — in joint space (top) and in cartesian hand space (bottom).

We can note a few distinct features of this mapping between joint and
hand space. First, equal spacing across the workspace in joint space
does *not* correspond to equal spacing across the hand workspace,
especially near the outer edges of the hand's reach. Second, a square
workspace region in joint space corresponds to a really curved region
in hand space. These complexities reflect the fact that the mapping
between joint space and hand space is *non-linear*.

### Inverse Kinematics

You can start to appreciate the sorts of problems the brain must face when planning arm movements. If I want move my hand through a particular hand path, what joint angles does that correspond to? We must use inverse kinematics to determine this.

I will leave the inverse kinematics equations up to you to derive as one of the steps in your assignment for this topic.

### The Jacobian

So far we have looked at kinematic equations for arm positions — joint angular positions and hand cartesian positions. Here we look at velocities (rate of change of position) in the two coordinate frames, joint-space and hand-space.

How can we compute hand velocities \(\frac{dH}{dt}\) given joint velocities \(\frac{d\theta}{dt}\)? If we can we compute an intermediate term \(\frac{dH}{d\theta}\) then we can apply the the Chain Rule:

\begin{equation} \frac{dH}{dt} = \frac{dH}{d\theta} \frac{d\theta}{dt} \end{equation}
This intermediate term is in fact known as the **Jacobian**
(Wikipedia:Jacobian) matrix \(J(\theta)\) and is defined as:

and so:

\begin{equation} \frac{dH}{dt} = J(\theta) \frac{d\theta}{dt} \end{equation}Note that the Jacobian is written as \(J(\theta)\) which means it is a function of joint angles \(\theta\) — in other words, the four terms in the Jacobian matrix (see below) change depending on limb configuration (joint angles). This means the relationships between joint angles and hand coordinates changes depending on where the limb is in its workspace. We already have an idea that this is true, from the figure above.

For the sake of notational brevity we will refer to hand velocities as \(\dot{H}\) and joint velocities as \(\dot{\theta}\) and we will omit (from the notation) the functional dependence of \(J\) on \(\theta\):

\begin{equation} \dot{H} = J \dot{\theta} \end{equation}The Jacobian is a matrix-valued function; you can think of it like a vector version of the derivative of a scalar. The Jacobian matrix \(J\) encodes relationships between changes in joint angles and changes in hand positions.

\begin{equation} J = \frac{dH}{d\theta} = \left[ \begin{array}{cc} \frac{\partial H_{x}}{\partial \theta_{1}}, &\frac{\partial H_{x}}{\partial \theta_{2}}\\ \frac{\partial H_{y}}{\partial \theta_{1}}, &\frac{\partial H_{y}}{\partial \theta_{2}} \end{array} \right] \end{equation}For a system with 2 joint angles \(\theta = (\theta_{1},\theta_{2})\) and 2 hand coordinates \(H = (H_{x},H_{y})\), the Jacobian \(J\) is a 2x2 matrix where each value codes the rate of change of a given hand coordinate with respect the given joint coordinate. So for example the term \(\frac{\partial H_{x}}{\partial \theta_{2}}\) represents the change in the \(x\) coordinate of the hand given a change in the elbow joint angle \(\theta_{2}\).

So how do we determine the four terms of \(J\)? We have to do calculus
and differentiate the equations for \(H_{x}\) and \(H_{y}\) with respect
to the joint angles \(\theta_{1}\) and \(\theta_{2}\). Fortunately we can
use a Python package for symbolic computation called SymPy to help us
with the calculus. (you will have to install SymPy… on Ubuntu (or
any Debian-based GNU/Linux), just type ```
sudo apt-get install
python-sympy
```

)

# import sympy from sympy import * # define these variables as symbolic (not numeric) a1,a2,l1,l2 = symbols('a1 a2 l1 l2') # forward kinematics for Hx and Hy hx = l1*cos(a1) + l2*cos(a1+a2) hy = l1*sin(a1) + l2*sin(a1+a2) # use sympy diff() to get partial derivatives for Jacobian matrix J11 = diff(hx,a1) J12 = diff(hx,a2) J21 = diff(hy,a1) J22 = diff(hy,a2) print J11 print J12 print J21 print J22

In [12]: print J11 -l1*sin(a1) - l2*sin(a1 + a2) In [13]: print J12 -l2*sin(a1 + a2) In [14]: print J21 l1*cos(a1) + l2*cos(a1 + a2) In [15]: print J22 l2*cos(a1 + a2)

So now we have the four terms of the Jacobian:

\begin{equation} J = \left[ \begin{array}{cc} -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{array} \right] \end{equation}and now we can write a Python function that returns the Jacobian, given a set of joint angles:

def jacobian(A,aparams): """ Given joint angles A=(a1,a2) returns the Jacobian matrix J(q) = dH/dA """ l1 = aparams['l1'] l2 = aparams['l2'] dHxdA1 = -l1*sin(A[0]) - l2*sin(A[0]+A[1]) dHxdA2 = -l2*sin(A[0]+A[1]) dHydA1 = l1*cos(A[0]) + l2*cos(A[0]+A[1]) dHydA2 = l2*cos(A[0]+A[1]) J = matrix([[dHxdA1,dHxdA2],[dHydA1,dHydA2]]) return J

and now we can use the Jacobian to compute hand velocities, given joint angular velocities according to the equation from above:

\begin{equation} \dot{H} = J \dot{\theta} \end{equation}aparams = {'l1' : 0.3384, 'l2' : 0.4554} A = array([45.0,90.0])*pi/180 # joint angles Ad = matrix([[-5.0],[3.0]])*pi/180 # joint velocities J = jacobian(A,aparams) Hd = J*Ad print Hd

[[ 0.03212204] [-0.00964106]]

We can visualize the Jacobian by plotting velocity vectors in joint
space and the corresponding velocity vectors in hand space. See source
file jacobian\_{plots.py} for python code.

Figure 5: Visualizing the Jacobian

- Accelerations
We can also use the Jacobian to compute accelerations at the hand due to accelerations at the joint, simply by differentiating our equations with respect to time. Remember the equation for velocity:

\begin{equation} \dot{H} = J \dot{\theta} \end{equation}for acceleration we just differentiate both sides with respect to time:

\begin{equation} \ddot{H} = \frac{d}{dt} \left(\dot{H}\right) = \frac{d}{dt} \left( J \dot{\theta} \right) \end{equation}and apply the Product Rule from calculus:

\begin{equation} \frac{d}{dt} \left( J \dot{\theta} \right) = \left[ J \left( \frac{d}{dt} \dot{\theta} \right) \right] + \left[ \left( \frac{d}{dt} J\right) \dot{\theta} \right] \end{equation}simplifying further,

\begin{equation} \ddot{H} = (J) (\ddot{\theta}) + (\dot{J}) (\dot{\theta}) \end{equation}So if we have joint accelerations \(\ddot{\theta}\) and joint velocities \(\dot{\theta}\), all we need is the Jacobian \(J\) and the time derivative of the Jacobian \(\dot{J}\), and we can compute hand accelerations \(\ddot{H}\).

How do we get \(\dot{J}\)? Again we can use SymPy, as above, and we get the following: (I will just show the final Python function, not the SymPy part):

def jacobiand(A,Ad,aparams): """ Given joint angles A=(a1,a2) and velocities Ad=(a1d,a2d) returns the time derivative of the Jacobian matrix d/dt (J) """ l1 = aparams['l1'] l2 = aparams['l2'] Jd11 = -l1*cos(A[0])*Ad[0] - l2*(Ad[0] + Ad[1])*cos(A[0] + A[1]) Jd12 = -l2*(Ad[0] + Ad[1])*cos(A[0] + A[1]) Jd21 = -l1*sin(A[0])*Ad[0] - l2*(Ad[0] + Ad[1])*sin(A[0] + A[1]) Jd22 = -l2*(Ad[0] + Ad[1])*sin(A[0] + A[1]) Jd = matrix([[Jd11, Jd12],[Jd21, Jd22]]) return Jd

Note that the time derivative of the Jacobian, \(\dot{J}\), is a function of both joint angles \(\theta\) and joint velocities \(\dot{\theta}\).

### Inverse Jacobian

We have seen how to compute hand velocities and accelerations given joint angles, velocities and accelerations… but what about inverse kinematics? How do we compute joint velocities given hand velocities? What about joint accelerations given hand accelerations?

Recall for velocities that:

\begin{equation} J \dot{\theta} = \dot{H} \end{equation}To solve for \(\dot{\theta}\) we can simply multiply both sides of the equation by the matrix inverse of the Jacobian:

\begin{equation} \dot{\theta} = J^{-1}\dot{H} \end{equation}
In Numpy you can compute the inverse of a matrix using the `inv()`

function.

Similarly for accelerations, we can do the following:

\begin{equation} \ddot{\theta} = J^{-1} \left[ \ddot{H} - \left( (\dot{J}) (\dot{\theta}) \right) \right] \end{equation}## The Redundancy Problem

Notice something important (and unrealistic) about our simple
two-joint arm model. There is a one-to-one mapping between any two
joint angles \((\theta_{1},\theta_{2})\) and hand position
\((H_{x},H_{y})\). That is to say, a given hand position is *uniquely
defined* by a single set of joint angles.

This is of course convenient for us in a model, and indeed many empirical paradigms in sensory-motor neuroscience construct situations where this is true, so that it's easy to go between intrinsic and extrinsic coordinate frames.

This is not how it is in the real musculoskeletal system, of course,
where there is a *many-to-one mapping* from intrinsic to extrinsic
coordinates. The human arm has not just two mechanical *degrees of
freedom* (independent ways in which to move) but seven. The shoulder
is like a ball-socket joint, so it can rotate 3 ways (roll, pitch,
yaw). The elbow can rotate a single way (flexion/extension). The
forearm, because of the geometry of the radius and ulna bones, can
rotate one way (pronation/supination), and the wrist joint can rotate
two ways (flexion/extension, and radial/ulnar deviation). This is to
say nothing about shoulder translation (the shoulder joint itself can
be translated up/down and fwd/back) and the many, many degrees of
freedom of the fingers.

With 7 DOF at the joint level, and only three cartesian degrees of
freedom at the hand (the 3D position of the hand) we have 4 extra
DOF. This means that there is a 4-dimensional "null-space" where joint
rotations within that 4D null space *have no effect on 3D hand
position*. Another way of putting this is, there are an infinite
number of ways of configuring the 7 joints of the arm to reach a
single 3D hand position.

How does the CNS choose to plan and control movements with all of this
redundancy? This is known in the sensory-motor neuroscience literature
as the *Redundancy Problem*.

## Computational Models of Kinematics

### The Minimum-Jerk Hypothesis

One of the early computational models of arm movement kinematics was described by Tamar Flash and Neville Hogan. Tamar was a postdoc at MIT at the time, working with Neville Hogan, a Professor there (as well as with Emilio Bizzi, another Professor at MIT). Tamar is now a Professor at the Weizmann Institute of Science in Rehovot, Israel.

For a long time, researchers had noted striking regularities in the hand paths of multi-joint arm movements. Movements were smooth, with unimodal, (mostly) symmetric velocity profiles (so-called "bell-shaped" velocity profiles).

- Morasso, P. (1981). Spatial control of arm movements. Experimental Brain Research, 42(2), 223-227.
- Soechting, J. F., & Lacquaniti, F. (1981). Invariant characteristics of a pointing movement in man. The Journal of Neuroscience, 1(7), 710-720.
- Abend, W., Bizzi, E., & Morasso, P. (1982). Human arm trajectory formation. Brain: a journal of neurology, 105(Pt 2), 331.
- Atkeson, C. G., & Hollerbach, J. M. (1985). Kinematic features of unrestrained vertical arm movements. The Journal of Neuroscience, 5(9), 2318-2330.

Flash and Hogan investigated these patterns in the context of
optimization theory — a theory proposing that the brain plans and
controls movements in an *optimal* way, where optimal is defined by a
specific task-related cost function. In other words, the brain chooses
movement paths and neural control signals that *minimize* some
objective cost function.

- Todorov, E. (2004). Optimality principles in sensorimotor control. Nature neuroscience, 7(9), 907-915.
- Diedrichsen, J., Shadmehr, R., & Ivry, R. B. (2010). The coordination of movement: optimal feedback control and beyond. Trends in cognitive sciences, 14(1), 31-39.

Flash and Hogan wondered, at the kinematic level, what cost function might predict the empirically observed patterns of arm movements?

They discovered that by minimizing the time integral of the square of "jerk", a simple kinematic model predicted many of the regular patterns seen empirically for arm movements of many kinds (moving from one point to another, or even moving through via-points and moving around obstacles). Jerk is the rate of change (the derivative) of acceleration, i.e. the second derivative of velocity, or the third derivative of position. Essentially jerk is a measure of movement smoothness. Whether or not it turns out that the brain is actually interested in minimizing jerk in order to plan and control arm movements (an explanatory model), the minimum-jerk model turns out to be a good descriptive model that is able to predict kinematics of multi-joint movement.

- Hogan, N. (1984). An organizing principle for a class of voluntary movements. The Journal of Neuroscience, 4(11), 2745-2754.
- Flash, T. and Hogan, N. (1985) The coordination of arm movements: an experimentally confirmed mathematical model. J. Neurosci. 7: 1688-1703.

### Minimum Endpoint Variance

More recently, researchers have investigated how noise (variability) in neural control signals affects movement kinematics. One hypothesis stemming from this work is that the CNS plans and controls movements in such a way as to minimize the variance of the endpoint (e.g. the hand, for a point-to-point arm movement). The idea is that the effects of so-called "signal-dependent noise" in neural control signals accumulates over the course of a movement, and so the hypothesis is that the CNS chooses specific time-varying neural control signals that minimize the variability of the endpoint (e.g. hand), for example at the final target location of a point-to-point movement.

- Harris, C. M., & Wolpert, D. M. (1998). Signal-dependent noise determines motor planning. Nature, 394(6695), 780-784.
- van Beers, R. J., Haggard, P., & Wolpert, D. M. (2004). The role of execution noise in movement variability. Journal of Neurophysiology, 91(2), 1050-1063.
- Iguchi, N., Sakaguchi, Y., & Ishida, F. (2005). The minimum endpoint variance trajectory depends on the profile of the signal-dependent noise. Biological cybernetics, 92(4), 219-228.
- Churchland, M. M., Afshar, A., & Shenoy, K. V. (2006). A central source of movement variability. Neuron, 52(6), 1085-1096.
- Simmons, G., & Demiris, Y. (2006). Object grasping using the minimum variance model. Biological cybernetics, 94(5), 393-407.

## How does the brain plan and control movement?

Georgopoulos and colleagues in the 1980s recorded electrical activity of single motor cortical neurons in awake, behaving monkeys during two-joint arm movemnets. The general goal was to find out if patterns of neural activity could inform the question of how the brain controls arm movements, and in particular, how large populations of neurons might be coordinated to produce the patterns of arm movements empirically observed.

- Georgopoulos, A. P., Kalaska, J. F., Caminiti, R., & Massey, J. T. (1982). On the relations between the direction of two-dimensional arm movements and cell discharge in primate motor cortex. The Journal of Neuroscience, 2(11), 1527-1537.
- Georgopoulos, A. P., Caminiti, R., Kalaska, J. F., & Massey, J. T. (1983). Spatial coding of movement: a hypothesis concerning the coding of movement direction by motor cortical populations. Exp Brain Res Suppl, 7(32), 336.
- Georgopoulos, A. P., Schwartz, A. B., & Kettner, R. E. (1986). Neuronal population coding of movement direction. Science, 233(4771), 1416-1419.
- Schwartz, A. B., Kettner, R. E., & Georgopoulos, A. P. (1988). Primate motor cortex and free arm movements to visual targets in three-dimensional space. I. Relations between single cell discharge and direction of movement. The Journal of Neuroscience, 8(8), 2913-2927.
- Georgopoulos, A. P., Kettner, R. E., & Schwartz, A. B. (1988). Primate motor cortex and free arm movements to visual targets in three-dimensional space. II. Coding of the direction of movement by a neuronal population. The Journal of Neuroscience, 8(8), 2928-2937.
- Kettner, R. E., Schwartz, A. B., & Georgopoulos, A. P. (1988). Primate motor cortex and free arm movements to visual targets in three-dimensional space. III. Positional gradients and population coding of movement direction from various movement origins. The journal of Neuroscience, 8(8), 2938-2947.
- Moran, D. W., & Schwartz, A. B. (1999). Motor cortical representation of speed and direction during reaching. Journal of Neurophysiology, 82(5), 2676-2692.

### An Alternate View

A paper by Mussa-Ivaldi in 1988 showed that in fact the Georgopoulos finding that broadly tuned neurons seem to encode hand movement kinematics does not imply that the brain encodes kinematics… Mussa-Ivaldi showed that because in the motor system so many kinematic and dynamic variables are correlated, broadly tuned neurons in motor cortex could be shown to "encode" many intrinsic and extrinsic kinematic and dynamic variables.

- Mussa-Ivaldi, F. A. (1988). Do neurons in the motor cortex encode movement direction? An alternative hypothesis. Neuroscience Letters, 91(1), 106-111.

### Neural Prosthetics

The Georgopoulos results have been directly translated into the very applied problem of using brain activity to control neural prosthetics, e.g. robotic prosthetic limbs. Form a practical point of view, it may not matter how cortical neurons encode movement, as long as we can build computational models (even just statistical models) of the relationship between cortical activity and limb motion. Here are some examples.

- Chapin, J. K., Moxon, K. A., Markowitz, R. S., & Nicolelis, M. A. (1999). Real-time control of a robot arm using simultaneously recorded neurons in the motor cortex. Nature neuroscience, 2, 664-670.
- Taylor, D. M., Tillery, S. I. H., & Schwartz, A. B. (2002). Direct cortical control of 3D neuroprosthetic devices. Science, 296(5574), 1829-1832.
- Donoghue, J. P. (2002). Connecting cortex to machines: recent advances in brain interfaces. Nature Neuroscience, 5, 1085-1088.
- Wolpaw, J. R., & McFarland, D. J. (2004). Control of a two-dimensional movement signal by a noninvasive brain-computer interface in humans. Proceedings of the National Academy of Sciences of the United States of America, 101(51), 17849-17854.
- Musallam, S., Corneil, B. D., Greger, B., Scherberger, H., & Andersen, R. A. (2004). Cognitive control signals for neural prosthetics. Science, 305(5681), 258-262.
- Hochberg, L. R., Serruya, M. D., Friehs, G. M., Mukand, J. A., Saleh, M., Caplan, A. H., … & Donoghue, J. P. (2006). Neuronal ensemble control of prosthetic devices by a human with tetraplegia. Nature, 442(7099), 164-171.
- Santhanam, G., Ryu, S. I., Byron, M. Y., Afshar, A., & Shenoy, K. V. (2006). A high-performance brainâ€“computer interface. nature, 442(7099), 195-198.
- Velliste, M., Perel, S., Spalding, M. C., Whitford, A. S., & Schwartz, A. B. (2008). Cortical control of a prosthetic arm for self-feeding. Nature, 453(7198), 1098-1101.

## Why are kinematic transformations important?

The many-to-one mapping issue is directly relevant to current key
questions in sensory-motor neuroscience. How does the nervous system
choose a single arm configuration when the goal is to place the hand
at a specific 3D location in space? Of course the *redundancy problem*
as it's known, is not specific to kinematics. We have many more
muscles than joints, and so the problem crops up again: how does the
CNS choose a particular set of time-varying muscle forces, to produce
a given set of joint torques? The redundancy problem keeps getting
worse as we go up the pipe: there are *many* more neurons than
muscles, and again, how does the CNS coordinate millions of neurons to
control orders-of-magnitude fewer muscles? These are key questions
that are still unresolved in modern sensory-motor neuroscience. Making
computational models where we can explicitly investigate these
coordinate transformations, and make predictions based on different
proposed theories, is an important way to address these kinds of
questions.

Another category of scientific question where modeling kinematic
transformations comes in handy, is related to *noise* (I don't mean
acoustic noise, i.e. loud noises, but random variability). It is known
that neural signals are "noisy". The many transformations that sit in
between neuronal control signals to muscles, and resulting hand
motion, are complex and nonlinear. How do noisy control signals
manifest in muscle forces, or joint angles, or hand positions? Are
there predictable patterns of variability in arm movement that can be
attributed to these coordinate transformations? If so, can we study
how the CNS deals with this, i.e. compensates for it (or not)?

This gives you a flavour for the sorts of questions one can begin to address, when you have an explicit quantitative model of coordinate transformations. There are many studies in arm movement control, eye movements, locomotion, etc, that use this basic approach, combining experimental data with predictions from computational models.

- Scott, S. H., and G. E. Loeb. "The computation of position sense from spindles in mono-and multiarticular muscles." The Journal of neuroscience 14, no. 12 (1994): 7529-7540.
- Tweed, Douglas B., Thomas P. Haslwanter, Vera Happe, and Michael Fetter. "Non-commutativity in the brain." Nature 399, no. 6733 (1999): 261-263.
- Messier, J., & Kalaska, J. F. (1999). Comparison of variability of initial kinematics and endpoints of reaching movements. Experimental Brain Research, 125(2), 139-152.
- Loeb, E. P., S. F. Giszter, P. Saltiel and E. Bizzi, and F. A. Mussa-Ivaldi. "Output units of motor behavior: an experimental and modeling study." Journal of cognitive neuroscience 12, no. 1 (2000): 78-97.
- Selen, Luc PJ, David W. Franklin, and Daniel M. Wolpert. "Impedance control reduces instability that arises from motor noise." The Journal of Neuroscience 29, no. 40 (2009): 12606-12616.

## Next Steps

Next we will look at modelling the dynamics of arm movement.