1. Research seminar at NTNU, 27 September 2007
The BUMP model: Speed-accuracy tradeoffs and velocity profiles of aimed movement
Welcome
Have you got handouts?
My name, 3rd year PhD student UNSW
Here to talk about one part of my thesis (speed-acc and velocity profiles)‏
Contact details bottom
Robin T. Bye PhD student in neuroengineering School of Electrical Engineering & Telecommunications University of New South Wales Sydney Australia

2. Presentation outline What is neuroengineering?
Invariants in aimed movements
Speed-accuracy tradeoffs
Velocity profiles
The BUMP model
Simulation experiments
Summary
Q & A

3. What is neuroengineering?

4. What is neuroengineering?
Emerging interdisciplinary field of research
Electrical/computer engineering
Control systems, signal processing, neural networks, etc.
Neural tissue engineering
Computational/experimental neuroscience
Materials science
Clinical neurology
Nanotechnology
Other areas
My area of research is control system
Mention background (BE and MengSc)‏
Other areas are

5. What is neuroengineering?
Goal: “Reverse-engineer” the nervous system
How does it function?
How can we modify it?
What can we learn from the brain?
Bidirectional inspiration: humans vs. external world
Existing and potential human improvements inspired by external world:
Bionic limb
“Mind control” through brain-computer interface
From my perspective, the goal is
If the brain were an ipod...
One way: Create other appliances etc inspired by ipod
Other way: Fix ipod when broken

6. What is neuroengineering?
“Pacemaker” for cerebral palsy, stuttering, lesions
Cochlea implant
External world improvements inspired by the human CNS
Robotics and control systems
Information processing and coding algorithms, e.g.
neural networks
face recognitioning systems
Example of bionic arm presented next...
To illustrate what neuroengineering is all about...

7. Explain points 1-4.
I will now show you a video containing a woman with such a bionic arm. The video is from a news report on the American news television channel CNN.
Bionic arm schematic. Adapted from the Washington Post (2006).

8. Claudia Mitchell's bionic arm
Claudia Mitchell's bionic arm. Adapted from CNN report: Lady with bionic arm (2006).

9. Reverse-engineering the brain
The nervous system is a “black box”
Input (instruction)
Output (response) ?
System (CNS)
Cannot take the brain apart and put it together again!
Second-best:
Medical imaging, e.g. fMRI → “grey” box?
Given inputs, deduce black box system from outputs
I mentioned the ipod earlier, however, we cannot just open the brain and put it together again!
(although open-scull surgery has revealed certain things)‏
The use of non-invasive methods have exploded recently. Using medical imaging, possible to map functions of brain, e.g. fMRI. The black box is thus not completely black anymore.
Another way is to provide the black box with instructions, or inputs, and then observe the responses (outputs). I can press play on the ipod and it plays music, therefore there must be a music player inside. How does it work? I can choose between many songs, thus it has memory, and so on.

10. Reverse-engineering the brain
Control systems approach
Mathematical description of signals and systems
Applicable in modelling human movements
The CNS (black box) is a system
Movement instruction → response execution
E.g. lift a glass, say “Aaaah”, or move from A to B
Response may reveal properties of the CNS control system
Deviation from desired response?
Common characteristics across responses?
Popular approach in control systems theory
Explain a little about control systems
Inputs and outputs in human beings
Expand on points about deviation and characteristics
(e.g. time delay in reaction time, error in endpoint are deviations, velocity profile, straight-line trajectory is a characteristic)‏
Lead us to next topic of presentation: Invariants in aimed movements

11. Invariants in aimed movements

12. Invariants in aimed movements
Common response characteristics across subjects, tasks, and environments constitute invariants
E.g. increased movement variance with increased speed, single-peaked velocity profile, approximately straight line trajectory, response timing, etc.
Why do invariants occur?
Intrinsic properties, e.g. transmission times, biomechanical system, external world influences
Response planning strategies in CNS
Transmission times and the physiological properties of the biomechanical system will influence the response, for example by causing time delays.
More interested in central processing mechanisms that might play a role. Can these invariants results from response planning strategies?

13. Useful measures of aimed movements
Quantitative measures of aimed movements
Movement time T
Target distance D
Target width W
Endpoint error measures, e.g.
Absolute error |E| or square error E²
(why not signed (negative) errors?)
Standard deviation S or variance S²
Before I proceed further, I will talk about some useful measures when making movements from a point A to a point B and illustrate with a typical 1D movement task.

14. 1D movement task E4
Overshoot region E3 W
Centre of target E2
Undershoot region E1 m4
Four possible movements:
m1 misses target (too short)
m2 undershoots centre of target
m3 overshoots centre of target
m4 misses target (too long) m3 D
Typical 1D task where goal is to hit the center line target. Hence, the subject may either overshoot or undershoot target. Depending on the task paradigm, one might measure number of hits inside the target region (example of a serial task, often with two such target regions), or one may measure independent movements and their errors from the centre line. m2 m1
Start

15. 1D movement invariants in this presentation
Speed-accuracy tradeoffs
Logarithmic tradeoff (Fitts' law)
Linear tradeoff
Velocity profiles
Asymmetrical (left-skewed) profile
Symmetrical profile
This presentation focusses on such 1D movements presented above. I will now talk about two important invariants in such movements, namely speed-accuracy tradeoff and velocity profiles.
Log and Lin are the two most prominent speed-accuracy tradeoff
Velocity profiles may be asymmetrical or symmetrical. In terms of asymmetrical profiles I will talk about left-skewed asymmetrical profiles, although there are reports of right-skewed ones.

16. Speed-accuracy tradeoff
Facts from experiments and common experience:
Faster movement leads to less accuracy
High accuracy requires slower movement
If it exists, what is the mathematical function?
We intuitively know this is right.
Is there a mathematical function to describe this relationship?
f(x) = ?
Error
* For a fixed distance, speed is
inversely related to movement
time
Movement time*

17. Logarithmic tradeoff Reciprocal tapping experiment by Fitts (1954)
Observe T for fixed combinations of D and W
Count number of target hits during period of time
Move fast while hit rate at least 95%
Count hits inside black regions. If hit rate sank below 95%, dismiss data set. Emphasise speed (time-minimisation task).
Fitts' reciprocal tapping task. Adapted from Fitts (1954).

18. Fitts' logarithmic law T increases with greater target distance D
T increases with smaller target width W
Fitts' law: T = a + b log2(2W/D)
Index of difficulty: Id = log2(2W/D)
Linear form of Fitts' law : T = a + b Id
Both make intutive sense: More difficult to hit target further away or smaller.
Movement time T
bId a
Index of difficulty Id

19. Fitts' law A “Newton's law” for human movements?
Holds for extraordinary amount of paradigms:
People: Kids, adults, elderly, mentally challenged, drugged, ...
Manipulators: Joystick, mouse, keyboard, foot pedal, ...
Environments: On land, under water, in aircraft flights, ...
Other: Discrete movements, without visual feedback, vision through microscope, ...
However, fails for timed movements!
Inclusion of temporal goal → linear tradeoff
Fitt's law has proven extremely robust.
Fitts conducted another experiment ten years later using discrete movement, i.e. examining single-tapping movements and their endpoint errors and came to exactly the same logarithmic result.
Timed movement: You have to make the movement hit the target after a prespecified duration has elapsed, i.e. move from A to B in exactly 500 ms. If movement is too slow or too fast by a margin, say 10%, then discharge movement from data set.

20. Linear tradeoff Discrete tapping experiment, Schmidt et al. (1979)
≈ Fitts-like experiment + temporal goals (desired T)
Result: Standard deviation S of endpoint varies linearly with average movement speed D/T
Linear law: S = a + bD/T
Some years later Schmidt decided to include temporal constraints in a discrete tapping experiment.
Would collect results from movements with different movement speeds, that is, by combinations of D and T.
They only tried movements of movement durations T of 140, 170, 200 ms, however, the linear laws also shows up in slower movements, as has been shown in numerous experiments
Standard deviation S
bD/T a
Average movement speed D/T

21. Linear tradeoff Holds for variety of time-matching tasks, including
single tapping tasks
saccadic eye movements
wrist rotations
other time-matching tasks (see Zelaznik, 1993, for review)
Again, very strong relationship.
As long as the movement is timed, this relationship is likely to show up, no matter which limb or environment you are in.

22. Other tradeoffs? Many have been suggested:
Other logarithmic or linear laws
Power laws
Delta-lognormal law
Some may fit better for particular experiments
Sometimes “academic” improvement
Fitts' and Schmidt's laws de facto tradeoffs
Academic improvement: Slightly better line fittings etc. However, it is better to have a simple relationship explaining a lot, than a lot of complex relationships for different paradigms. Increasing number of parameters normally results in better line fitting, however, those parameters may not be usable in a different paradigm. Both Fitts' law and the linear law are extremely robust and de facto tradeoffs referred to.

23. Velocity profiles
Aimed movements usually have single-peak velocity profiles for
almost any limb
single- and multi-joint movements
different environments
different inertial loads
different movement speeds
target sizes, shapes, and distances (see Plamondon, 1997, for review)
Velocity profile is a plot of how velocity changes during the course of movement. The next slides will give you an idea.

24. Velocity profiles Symmetrical profiles
ballistic movements (≈ 100 ms duration)
movements with temporal goals
Velocity
Velocity
Ballistic movements are the fastest movements we can make, with a duration of approximately 100 ms. They are ballistic, as there is no feedback involved and no error-correction is possible. Such profiles are typically symmetrical, i.e. max velocity occurs midway through movement.
In addition, timed movements are usually also symmetrical. 50
100
150
300
Time (ms)
Time (ms)
Ballistic movement
300 ms timed movement

25. Velocity profiles Asymmetrical (left-skewed) profiles
Non-ballistic movements incorporating feedback
Movements with spatial constraints only
Skewness increases with movement time (Beggs & Howarth, 1972)
Velocity
Velocity
Left-skewed profiles occur in slower than ballistic movements where feedback is involved. Left-skewed means that the max velocity occurs early in the movement.
We typically see these movements in Fitts-like task with spatial constraints such as minimise endpoint error.
In such cases, subjects will usually move fast initially, and then zoom in on the target more slowly.
Has been shown that this skewness increases with movement time, i.e. for slower movements.
There are reports of right-skewed profiles occuring in very fast movements. Not discussed here.
Time (ms)
150
Time (ms)
300
150 ms movement
300 ms movement

26. Summary Spatially constrained movements
The goal is to minimise endpoint error
Results in logarithmic speed-accuracy tradeoff (Fitts' law)
Results in asymmetrical (left-skewed) velocity profiles
Spatially + temporally constrained movements
The goal is to minimise endpoint error and make movement in prespecified duration
Results in linear speed-accuracy tradeoff
Results in symmetrical velocity profiles

27. The BUMP model

28. Modelling movement Why make models? Imitate human movements
Improve robotic applications
Predict human behaviour
Extend knowledge about CNS
Model consistent with human data?
If so, provides explanation of how the CNS may operate (if it is biologically-feasible)
If not, proves how the CNS may not work!
Nice to imitate human beings, but to me, it is more important to find out how the CNS works. Compare predictions of a model with real human data. If model is based on biologically-feasible hypotheses, the model can provide one particular explanation.
If the model fails, the result is not without value. Now we know one way the CNS does not work!

29. Some influential models
Deterministic iterative-corrections model (Crossman & Goodeve, 1963)
Impulse-variability model (Schmidt et al., 1979)
Minimum jerk model (Flash & Hogan, 1985)
Stochastic optimised-submovement model (Meyer et al., 1988)
Minimum torque-change model (Uno et al., 1989)
I have listed some influential models. Many more exist. In the case of the BUMP model, the deterministic iterative-corrections model, the impulse-variability model, and the stochastic optimised-submovement model are models which have elements in common with the BUMP model.
Common for these models is that they make little or no hypotheses about the functioning of the CNS. They mainly show that their model can reproduce similar results as human beings under various movement paradigms.
I will not talk much about such hypotheses myself in this presentation, however, publications from our laboratory have shown how the various components of our simulator are consistent with current knowledge of CNS function and structure. Our theoretical framework is called Adaptive model theory.

30. Adaptive model theory (AMT)
The BUMP model is part of AMT
Neuroengineering account of movement control
Fusion of adaptive control theory and neuroscience
Addresses major human movement science issues
e.g. intermittency, redundancy, resources, nonlinear interactions (see Neilson & Neilson, 2005, for review)
Three systems for information processing
Biologically-feasible neural network solution
AMT is a comprehensive neuroengineering account of movement control
Recommend 2005 paper
As mentioned before, biologically-feasible
Central to theory are the three processing systems

31. Three processing systems
Sensory analysis (SA) system
Response planning (RP) system (this presentation)
Response execution (RE) system
Operate independently and in parallel: The CNS can simultaneously
Plan appropriate response to a stimulus (RP system)
Execute response to an earlier stimulus (RE system)
Detect and store a subsequent stimulus (SA system)
Numerous experiments regarding double stimulus reaction time experiments have shown that they operate independently and in parallel.

32. Intermittency SA and RE systems operate continuously
RP system operates intermittently
System is refractory while operating on “chunks” of info
Fixed planning time interval to plan a response trajectory
Planning time interval Tp = 100 ms
Leads to repeating SA-RP-RE sequences: BUMPs
Movement consists of concatenated submovements
Each submovement has a fixed duration of 100 ms
Central to our theory is the concept of intermittency. We hypothesise that the SA and RE are continuous, while the RP system operates intermittently.
Directly related to the psychological refractory period, PRP. The PR system is refractory while working on a chunk of information, that is, incoming signals will need to wait for the RP system to become available.
Will show an illustration to explain.

33. Basic Unit of Motor Production (BUMP)

34. Response planning system
Planning in terms of sensory consequences
E.g. in an airplane, the pilot plans in terms of the consequences of moving the joystick rather than the hand movements controlling the joystick
Redundancy problem
Infinite trajectories to move from A to B
which one to choose?
Yet, trajectories usually have invariants
E.g. straight-line trajectory, single-peaked velocity profile
We hypothesise planning in terms of sensory consequences.
Redundancy: Infinite number of trajectories, yet, we seem to be using the same trajectory, or set of trajectories, every time! What I mean by this is that we tend to make movements that...
Question is: How does the CNS choose a particular trajectory?

35. Response selection
Adding constraints to a movement task limits possible trajectories
Optimal control: use a cost function for trajectories
Choose particular trajectory that minimises cost
Common cost functions:
Movement time
Movement distance or its derivatives
velocity, acceleration (energy), jerk, snap
Torque or torque-change
One solution to “get rid” of a number of trajectories is to create some constraints, e.g. do not move in a zigzag pattern or a semicircle but use a straight line.
In optimal control: Such constraints are cost functions.
Mathematical functions. The particular trajectory that minimises the cost is chosen. One example is minimise energy.
In a predominantly inertial system, e.g. the arm is much like a spring-mass system, minimising energy is equivalent to minimising acceleration.
Below are a few possible cost functions. In the BUMP model, we use acceleration.

36. Minimum acceleration approach
Choosing acceleration as cost criterion to minimise results in
minimum acceleration/energy trajectory
optimally smooth trajectories
trajectories that are S-shaped
symmetrical velocity profiles (peak half-way)
Rationale:
Equivalent to minimising metabolic energy
Computationally easier than jerk
By choosing acceleration at cost criterion, you get the following properties of the trajectory (list them)‏
It makes evolutionary sense to minimise energy, just think about a lion trying to capture a zebra. Both of them would not like to waste energy when trying to catch or escape the other.
Computationally easier: In our laboratory we have shown that tracking deteriorates if jerk is used as a cost function, simply because it is harder to predict. Think about it, if you watch something moving guessing its speed might not be that hard, and you might be able to tell if it is accelerating or decelerating but that is about it. Now, tell the jerk, the derivative of acceleration, is even harder.

37. Planning in accelerated time
Optimal trajectory R* generated every Tp = 100 ms
Duration of R* may be of much longer duration!
Optimal S-shaped trajectory R* with
500 ms duration. The trajectory moves
the response from a standstill position
at zero to a standstill position at unity.
During movement, only the first 100 ms
are executed. Then, an updated R*
replaces the old one. Again, only 100 ms
are executed. This series of submovements
repeats until the target is reached.
I have talked about intermittency and generation of optimal trajectories. Our hypothesis about intermittency implies that a new optimal trajectory is being planned at RP intervals, i.e. 100 ms. However, the duration of these trajectories may be far longer!
Here is an optimal minimum acceleration trajectory of duration 500 ms. It is S-shaped with a symmetrical velocity profile. However, after 100 ms of this trajectory, or submovement, has been executed, a new trajectory is being planned to reach the target. I will explain this next.
Optimal S-shaped trajectory

38. Variable horizon control
Duration of R* is called prediction horizon
Variable horizon control = ability to vary duration of R* at RP intervals
Strategies for varying the horizon:
Receding horizon control
Fixed horizon control
Others may exist
The initial and final states consist of values of position and velocity. When planning an R*, the RP system must predict the initial and final state. After executing one submovement, the response has a new position and velocity. From this state, another submovement will start. I will illustrate this on the next slide.
Can change the duration of R*. The RP system may start with a 500 ms trajectory for the first submovement, a and change it to a 400 ms for the next one, say.
This ability gives rise to the concept of variable horizon control.

39. Receding horizon control
The duration of R* remains constant
The prediction horizon recedes when approached
Let's say the RP sys plans to move from position zero to position 1 by using 300 ms trajectories. However, it only executes the first 100 ms of this trajectory. While executing, it predicts where the response (e.g. the hand) will be at the end of this submovement, and plans another submovement to reach position 1. Again, the duration of this submovement is being planned with a duration of 300 ms.

40. Fixed horizon control
R* planned to a point ahead fixed in time and space
The prediction horizon decreases as the fixed horizon is approached
I will talk about simulation results later, but you may have guessed already that these two control strategies are able to reproduce the speed-accuracy tradeoffs and velocity profiles talked about earlier. A guess about receding horizon control might not be that intuitive, but what about the fixed horizon control here? What can you say about the movement time from the diagram?
ANS: The target will be reached after 3 Tp, i.e. 300 ms.
This is a timed movement, and hence, this strategy should lead to a linear tradeoff and symmetrical velocity profiles. We will see if it does when I present the simulation results.

41. Inaccuracies in movement
Movements never perfect - why?
Inaccurate internal models of
external system (joystick, bicycle)
muscle control system, biomechanical system
Noise in the CNS
Broadband signal-dependent noise
Standard deviation increases with size of motor command
Remedy: Intermittent error corrections
One thing I have not mentioned is inaccuracies in movement. Why never perfect movement? Why errors?
Inaccurate models: E.g. CP patients may be able to make correct oral commands about how to respond, indicating a correct model of the external world, but may be unable to make their hand do the appropriate movement, indicating errors in models of the internal systems.
Noise: Broad agreement that there is noise in the CNS, and this noise exists as broadband signal-dependent noise. This means that the standard deviation increases with the size of the motor command.
One way to remedy errors during the course of movement is intermittent error corrections.

42. Intermittent error corrections
Receding horizon control.
Each optimal R* has a
duration of 100 ms.
Ri* = desired response
Ri = actual response
Ei = error (undershoots)‏
Note: Errors can equally
well overshoot target.
Noise causes the first submovement to undershoot the target. Note that error is highly exaggerated for illustration purposes and similary, note that errors can equally well overshoot.
The RP system predicted the response to hit the target in 100 ms, thus there is no movement during RE2. Then it detects that it has not hit the target after all, and a new R* is planned and executed. The error is reduced during this feedback process.

43. Simulation experiments

44. Simulator description
Implemented using MATLAB and Simulink
Every component is biologically-feasible
Simulations of step movements (discrete point-to- point 1D movements) employing variable horizon control
Receding horizon control
Fixed horizon control
Stochastic noise added to motor commands
Just a few words on the simulator without getting too technical.

45. Simulation results Receding horizon control, 500 ms movement
Top graph shows endpoint absolute error vs movement time. Error decreases exponentially with MT.
Bottom graph is the more commonly used equivalent graph, namely MT vs. ID. This shows that a higher ID requires longer MT.
Very good data fit for the solid lines, strong result.
Logarithmic speed-accuracy tradeoff

46. Simulation results Receding horizon control, 500 ms movement
I mentioned before that spatially constrained movements follow Fitts' law, but also have left-skewed velocity profiles. Here is a velocity profile for a 500 ms movement.
Asymmetrical (left-skewed) velocity profile

47. Simulation results Fixed horizon control, 100-500 ms movements
This graph shows the endpoint standard deviation versus the average velocity for a number of movements with durations from 100 to 500 ms.
The deviation increases linearly with the average velocity as depicted by solid line. Strong result.
Linear speed-accuracy tradeoff

48. Simulation results Fixed horizon control, 500 ms movement
I mentioned that timed movements have symmetrical velocity profiles. Here is a simulation of a 500 ms movement, which has a perfectly symmetrical velocity profile.
Symmetrical velocity profile

49. Receding horizon control
Logarithmic tradeoff: Goodness of fit
Coefficient of determination R² as a measure of goodness of fit for the best fit exponential and linear functions W = D × 2−λt and T = aId + b, respectively, for 10 cm step movements employing receding horizon control and prediction horizons Th = {100, 200, , 1000} ms.
I will just briefly show you tables of the results. Here are the speed-accuracy results for receding horizon control in terms of goodness of fit, i.e. how much of the data points can be explained by the mathematical function used, in this case, Fitts' law.
Strong results, all R² values higher than 0,98 for the linear form of Fitts' law using the index of difficulty.
Table of goodness of fit

50. Receding horizon control
Asymmetrical profile: Level of asymmetry
Level of asymmetry in velocity profiles for 10 cm step movements using receding horizon control given by the ratio of duration of positive and negative acceleration.
This table shows how the asymmetry ratio increases for slower movements using receding horizon control. The fastest ballistic movement is symmetrical, then the max velocity peak becomes more and more left-skewed as the prediction horizon is increased, i.e. slower movements are being made.
This is in line with human data.
Table of levels of asymmetry

51. Fixed horizon control Linear tradeoff: Goodness of fit
Groups of fixed horizon control step movements of varying initial prediction horizons and movement distances and their corresponding correlation coefficient R2 as a measure of goodness of fit for the best linear function We = a D/T + b. Th is the initial prediction horizon; D is the movement distance; and R2 is the correlation coefficient.
This table shows the goodness of fit for various groups movements consisting of combinations of movement distance and movement duration. The R² value is higher than 0,99 in all cases, i.e. a linear law can explain more than 99% of the data points.
In terms of the velocity profiles, all were symmetrical, i.e. the asymmetry ratio was 1, for all simulations. This matches human data.
Table of goodness of fit
Symmetrical profile: Asymmetry ratio is 1 for all cases, i.e. all velocity profiles are symmetrical.

52. Conclusions
Simulation results closely match observations in human movement experiments
Receding horizon control successfully reproduces
Logarithmic speed-accuracy tradeoff (Fitts' law)
Asymmetrical (left-skewed) velocity profiles
Linear speed-accuracy tradeoff
Symmetrical velocity profiles

53. Conclusions
Results strongly support the BUMP model and its underlying hypotheses about human motor control
The BUMP model provides a unique theoretical bridge between seemingly disparate speed-accuracty tradeoff and velocity profiles

54. Summary

55. Neuroengineering is about reverse-engineering the brain
One method: Create models and see if they match the real world
The model must be biologically-feasible
Then a successful model provides a possible solution
If unsuccessful, at least one theory is eliminated
Mere line-fitting is of little value

56. Spatially constrained movements
Goal: minimise error
Results:
Logarithmic speed-accuracy tradeoff (Fitts' law)
Asymmetrical (left-skewed) velocity profiles
Spatially and temporally constrained movements
Goal: minimise error & move on time
Linear speed-accuracy tradeoff
Symmetrical velocity profiles

57. The BUMP model Simulation results
Intermittent response planning → submovements
Suggests two different response planning strategies
Receding horizon control
Fixed horizon control
Simulation results
Receding horizon control reproduces
Logarithmic tradeoff
asymmetrical profiles
Fixed horizon control reproduces
Linear tradeoff
symmetrical profiles

58. The BUMP model provides one possible account of human aimed movements
Biologically-feasible
All components are based on existing structures and knowledge about the CNS
Unique, as it explains both important tradeoffs and corresponding velocity profiles within one theoretical framework
As far as I know, most models can only reproduce either the linear or the logarithmic tradeoff. The stochastic optimised-submovements model by Meyer may be an exception, but only for very fast movements.

59. Q & A
Thanks to you all for listening. Please feel free to ask me any questions now, or after the seminar, and I am also reachable by the address I gave you on the first slide.

60. References
Bionic arm schematic. The Washington Post, 14/09/2006, accessed from dyn/content/graphic/2006/09/14/GR html?referrer= link, on 7/09/2007.
Claudia Mitchell's bionic arm. CNN Report: Lady with experimental bionic arm, broadcast 14/09/2006, accessed from on 19/09/2007.
Fitts, P. M. (1954). The information capacity of the human motor system in controlling the amplitude of movement. Journal of Experimental Psychology, 47, 381–391.
Schmidt, R. A., Zelaznik, H., Hawkins, B., Frank, J. S., & Quinn, J. (1979). Motor-output variability: A theory for the accuracy of rapid motor acts. Psychological Review, 86 (5), 415–451.
Zelaznik, H. N. (1993). Necessary and sufficient conditions for the production of linear speed-accuracy trade-offs in aimed hand movements. In K. M. Newell & D. M. Corcos (Eds.), Variability and motor control (pp. 91–115). Human Kinetics Publishers.
Plamondon, R., & Alimi, A. M. (1997). Speed/accuracy trade-offs in target-directed movements. Behavioral and Brain Sciences, 20, 279– 349.
Crossman, E. R. F. W., & Goodeve, P. J. (1963/1983). Feedback control of hand-movement and Fitts’ law. Quarterly Journal of Experimental Psychology, 35A, 251–278. (Reprint of Communication to the Experimental Society (1963))‏
Flash, T., & Hogan, N. (1985). The coordination of arm movements: An experimentally confirmed mathematical model. Journal of Neuroscience, 5 (7), 1688–1703.
Meyer, D. E., Abrams, R. A., Kornblum, S., Wright, C. E., & Smith, J. E. K. (1988). Optimality in human motor performance: Ideal control of rapid aimed movements. Psychological Review, 95 (3), 340–370.
Uno, Y., Kawato, M., & Suzuki, R. (1989). Formation and control of optimal trajectory in human multijoint arm movement: Minimum torque-change model. Biological Cybernetics, 61, 89–101.
Neilson, P. D., & Neilson, M. D. (2005). An overview of adaptive model theory: solving the problems of redundancy, resources, and nonlinear interactions in human movement control. Journal of Neural Engineering, 2 (3), S279–S312.