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Infant grasp learning: a computational model

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Abstract

This paper presents ILGM (the Infant Learning to Grasp Model), the first computational model of infant grasp learning that is constrained by the infant motor development literature. By grasp learning we mean learning how to make motor plans in response to sensory stimuli such that open-loop execution of the plan leads to a successful grasp. The open-loop assumption is justified by the behavioral evidence that early grasping is based on open-loop control rather than on-line visual feedback. Key elements of the infancy period, namely elementary motor schemas, the exploratory nature of infant motor interaction, and inherent motor variability are captured in the model. In particular we show, through computational modeling, how an existing behavior (reaching) yields a more complex behavior (grasping) through interactive goal-directed trial and error learning. Our study focuses on how the infant learns to generate grasps that match the affordances presented by objects in the environment. ILGM was designed to learn execution parameters for controlling the hand movement as well as for modulating the reach to provide a successful grasp matching the target object affordance. Moreover, ILGM produces testable predictions regarding infant motor learning processes and poses new questions to experimentalists.

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Notes

  1. Technically, REINFORCE requires the firing probability of neurons be specified as a differentiable function of the input. Our algorithm does not conform to this criterion (i.e., Step 1) but it can be shown that a softmax approximation to Step 1 yields a learning rule similar to the one we used.

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Acknowledgements

This work was supported in part by a Human Frontier Science Program grant to MAA, which made possible useful discussions with Giacomo Rizzolatti and Vittorio Gallese, to whom we express our gratitude. The work of MAA was supported in part by grant DP0210118 from the Australian Research Council (R.A. Owens, Chief Investigator). The work of EO was supported in part by ATR Computational Neuroscience Laboratories, Kyoto, Japan.

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Correspondence to Erhan Oztop.

Appendices

Appendix 1

The simulation loop

The global logic of a simulation session is given below. This applies to all simulations except Simulation 1, where the function of the Wrist Rotation layer output is replaced by the automatic palm orientation described in Appendix 2. The Affordance layer encodes the orientation of the target object in SE3b, and the location of the object in SE4. In the remaining simulations it encodes the existence of a target object.

Step 1::

Encode the affordances presented to the circuit in Affordance layer (A). Set the vector o as the center of the target object.

Step 2::

Compute the outputs of Virtual Finger (V), Hand Position (H) and Hand Rotation (R) layers.

Step 3::

Generate movement parameters v, h, r.

Step 4::

Generate reach using the parameters v, h and r and o while monitoring for contact.

Step 5::

On contact (or failure to contact) compute a contact list.

Step 6::

Compute stability and generate reward signal rs based on the contact list.

Step 7::

Adapt weights using rs.

Step 8::

Go to Step 2, unless interrupted by the user.

Hand Position layer output encoding

The Hand Position layer generates a vector composed of azimuth, elevation and radius (α, β, r). The vector is related to the rectangular coordinates (in a left-handed coordinate system) as follows:

$$ x = r\cos \beta \sin \alpha ,\;y = r\sin \beta ,\;z = - \cos \beta \cos \alpha $$
(17)

Figure 14 illustrates the conversion graphically.

Fig. 14
figure 14

The conventions used in the spherical representation of the Hand Position layer output. Note that azimuth values of 180 and −180° coincide

Appendix 2

The arm/hand model

Since the arm/hand model we used in our simulations is a kinematics one, the absolute values of the modeled lengths are irrelevant and thus not specified here; however, the relative sizes of the segments are as shown in Fig. 15.

Fig. 15
figure 15

The arm/hand model has 19 DOFs as shown

The arm is modeled to have a 3DOF joint at the shoulder to mimic the human shoulder ball joint and a 1DOF joint in the elbow for lower arm extension/flexion movements (see Fig. 15). The wrist is modeled to have 3DOFs to account for extension/flexion, pronation/supination, and ulnar and radial deviation movements of the hand. Each finger except the thumb is modeled to have 2DOFs to simulate metacarpophalangeal and distalinterphalangeal joints of human hand. The thumb is modeled to have 3DOFs, one for the metacarpophalangeal joint and the remaining two for extension/flexion and ulnar and radial extension movements of the thumb (i.e., for the carpometacarpal joint of the thumb).

Inverse kinematics

The simulator coordinate system for the forward and inverse kinematics is left-handed. The zero posture and the effect of positive rotations of the arm joints are shown in Fig. 16.

Fig. 16
figure 16

Zero posture of the arm/hand together with arm DOFs is shown. The arrows indicate the positive rotations

When we mention the reach component, we imply the computation of trajectories of the arm joint angles (θ1, θ2,θ3,θ4) to achieve a desired position for the end effector (i.e. the inverse kinematics problem). The end effector could be any point on the hand (e.g., the wrist, index finger tip, middle finger joints, etc.) as long as it is fixed with respect to the arm and the length of the lower limb of the arm is extended (for the sake of kinematics computations) to account for the end effector position. The end effector used in the simulations was the tip of either the index or the middle finger (see Appendix 3). The forward mapping, F, of the kinematics chain is a vector-valued function of the joint angles relating the joint angles to the end-effector position.

The Jacobian transpose method for inverse kinematics can be derived as a gradient descent algorithm by minimizing the square of the distance between the current (p) and the desired end-effector position (p desired ). The key to the algorithm is a special matrix called the geometric Jacobian matrix (J), which relates end-effector Cartesian velocity to the angular velocities of the arm joints (Sciavicco and Siciliano 2000):

$$\left[ {\matrix{ {\dot x} \cr {\dot y} \cr {\dot z} \cr } } \right] = J\left[ {\matrix{ {\dot \theta _1 } \cr {\dot \theta _2 } \cr {\dot \theta _3 } \cr {\dot \theta _4 } \cr } } \right]$$
(18)

or in vector notation \( \dot{p} = J\dot{\theta } \).

Representing the upper arm length and the (extended) lower arm length with l 1 and l 2 , respectively; and abbreviating sin(θ 1 ) and cos(θ 1 ) with s 1 and c 1 ; sin(θ 2 ) and cos(θ 2 ) with s 2 and c 2 ; sin(θ 3 ) and cos(θ 3 ) with s 3 and c 3 ; sin(θ 4 ) and cos(θ 4 ) with s 4 and c 4 , the Jacobian matrix of our arm model can be written as:

$$ \begin{array}{*{20}l} {J \hfill} & { = \hfill} & {{\left[ {\begin{array}{*{20}c} {0} & {{l_{2} {\left( { - c_{2} s_{4} + s_{2} s_{3} c_{4} } \right)} - l_{1} c_{2} }} \\ {{l_{2} {\left( { - s_{1} c_{2} s_{4} - {\left( {s_{1} s_{2} s_{3} c_{4} + c_{1} c_{3} c_{4} } \right)}} \right)} + l_{1} s_{1} c_{2} }} & {{l_{2} {\left( {c_{1} s_{2} s_{4} + c_{2} s_{3} c_{1} c_{4} } \right)} + l_{1} c_{1} c_{2} }} \\ {{l_{2} {\left( { - c_{1} c_{2} s_{4} + c_{1} c_{4} s_{2} s_{3} - c_{3} c_{4} s_{1} } \right)} - l_{1} c_{1} c_{2} }} & {{l_{2} {\left( { - s_{1} s_{2} s_{4} + c_{2} c_{4} s_{1} s_{3} } \right)} + l_{1} s_{1} s_{2} }} \\ \end{array} } \right.} \hfill} \\ {{} \hfill} & {{} \hfill} & {{} \hfill} \\ {{} \hfill} & {{} \hfill} & {{\left. {\begin{array}{*{20}c} {{ - l_{2} {\left( {c_{2} c_{3} c_{4} } \right)}}} & {{l_{2} {\left( { - s_{2} c_{4} + c_{2} s_{3} s_{4} } \right)}}} \\ {{l_{2} {\left( {c_{1} s_{2} c_{3} c_{4} + s_{1} s_{3} c_{4} } \right)}}} & {{ - l_{2} {\left( { - c_{1} c_{2} c_{4} - c_{1} s_{2} s_{3} s_{4} + s_{1} c_{3} s_{4} } \right)}}} \\ {{l_{2} {\left( { - c_{3} c_{4} s_{1} s_{2} - c_{1} c_{4} s_{3} } \right)}}} & {{l_{2} {\left( { - s_{1} c_{2} c_{4} - s_{1} s_{2} s_{3} s_{4} - c_{1} c_{3} s_{4} } \right)} - l_{1} c_{1} c_{2} }} \\ \end{array} } \right]} \hfill} \\ \end{array} $$
(19)

Then the algorithm is simply composed of iterating the following update rule until p=p desired , where η represents the joint angle update rate.

$$ \theta _{{t + 1}} = \theta _{t} + \eta J^{T}_{{\theta _{t} }} (p_{{desired}} - p_{t} ) $$
(20)

Note that the time dependency of the variables and the dependency of the Jacobian matrix on the joint angles are indicated with subscripts.

Automatic hand orientation

The automatic hand orientation employed in SE1a is modeled as minimizing the angle between the palm normal (X) and the vector (d) connecting the center of the object to the index finger knuckle using the hand’s extension/flexion degree of freedom (see Fig. 17). The angle is minimized when the palm normal coincides with the projection of d on to the extension/flexion plane of the hand.

Fig. 17
figure 17

The automatic hand orientation is modeled as minimizing the angle between the vectors X (palm normal) and d

When the hand makes a rotation of φ radians as illustrated in Fig. 17, the palm normal coincides with the projection of d on to the extension/flexion plane of the hand. Noting that object, index, pinky, wrist and elbow in Fig. 17 represent three-dimensional position vectors, the angle φ can be obtained as follows:

$$\eqalign{ & X = {{(pinky - index) \otimes (wrist - index)} \over {\left| {(pinky - index) \otimes (wrist - index)} \right|}} \cr & Y = {{wrist - elbow} \over {\left| {wrist - elbow} \right|}} \cr} $$
(21)
$$\eqalign{ & d = object - index \cr & d_{prj} = \langle X \cdot d\rangle X + \langle Y \cdot d\rangle Y \cr} $$
(22)
$$\varphi = - \cos ^{ - 1} \left( {{{\langle d_{prj} \cdot X\rangle } \over {\left| {d_{prj} } \right|}}} \right)$$
(23)

When automatic hand orientation is engaged, the hand is rotated by φ radians at each cycle of the simulation while reach is taking place. Note that when d prj is zero, the angle φ is not defined. In that case, the angle is returned as zero (i.e., hand is not rotated). This situation happens when the extension/flexion movement of the hand has no effect on the angle between the palm normal and d; in other words, when d is vertical to both X and Y.

Appendix 3

ILGM simulation parameters

The main behavior of the simulation system is determined by a resource file where many simulation parameters can be set. In this file, the three-dimensional positions and vectors are defined using a spherical coordinate system. The PAR, MER and RAD tags are used to indicate elevation, azimuth and radius components, respectively.

Object axis orientation range parameters

Object axis orientation range parameters define the minimum and maximum allowed tilt of the object around the z-axis (in the frontal plane). Ten units are allocated for encoding the tilt amount. These parameters are only used in SE3.

$$ \begin{array}{*{20}l} {{{\text{minTILT}}} \hfill} & {{\text{0}} \hfill} \\ {{{\text{maxTILT}}} \hfill} & {{{\text{90}}} \hfill} \\ \end{array} $$
(24)

Base learning rate parameter

Base learning rate parameter is used as the common multiplier for all the learning rates in the grasp learning circuit as η AR  = η VR  = η HR   = eta; η AV  = η AH   = η/MAXROTATE.

$$ eta\;(\eta )\;0.5{\text{ }} $$
(25)

LGM layer size parameters

LGM layer size parameters define the number of units to allocate for the layers generating the motor parameters. The Virtual finger (V) layer is composed of ten units which specify the synergistic control of the fingers. In what follows BANK, PITCH and HEADING tags indicate the supination-pronation, wrist extension-flexion and radial/ulnar deviation movements, respectively. The size of the Wrist Rotation (R) layer is determined with the following parameters (in this example 9×9×1 units will be allocated):

$$\eqalign{ & {\rm hand\_rotBANK\_code\_len } \;\quad\quad\quad {\rm 9 } \cr & {\rm hand\_rotPITCH\_code\_len } \quad\quad\quad {\rm 9} \cr & {\rm hand\_rotHEADING\_code\_len } \quad {\rm 1} \cr} $$
(26)

In what follows, the tags locMER, locPAR and locRAD indicate the Hand Position (H) layer components. The size of the Hand Position layer is determined with the following parameters (in this example 7×7×1 units will be allocated)

$$\eqalign{ & {\rm hand\_locMER\_code\_len } \quad {\rm 7} \cr & {\rm hand\_locPAR\_code\_len } \;\quad {\rm 7} \cr & {\rm hand\_locRAD\_code\_len } \quad {\rm 1} \cr} $$
(27)

Learning session parameters

Learning session parameters define the behavior of the simulator during learning. For a learning session, the simulator makes MAXBABBLE number of reach/grasp attempts. For each approach-direction (H layer output), the simulator makes MAXROTATE grasping attempts. After MAXREACH reaches are done, the next input condition is selected (e.g., the object orientation is changed). MAXBABBLE limits the maximum number of attempts the simulator will make. A particular simulation may be stopped at any instant. The saved connection weights then can be used for testing the performance later. Reach2Target parameter indicates which part of the hand should be used as the end effector by MG module for reach execution. The possible values are [INDEX, MIDDLE, THUMB] × [0,1,2] where 2 indicates the tip and 0 indicates the knuckle. An example set of parameter specification is as follows:

$$\eqalign{ & {\rm MAXREACH } \;\;\quad\quad\;\;\; {\rm 5 } \cr & {\rm MAXROTATE } \quad\quad\;\;\; {\rm 7 } \cr & {\rm MAXBABBLE } \quad {\rm 10000 } \cr & {\rm weightSave } \quad\quad\quad\; {\rm 4500 } \cr & {\rm Reach2Target } \quad {\rm INDEX1 } \cr} $$
(28)

Grasp stability parameters

Grasp stability parameters define the acceptable grasps in terms of physical stability. costThreshold specifies the allowable inaccuracy in grasping. Ideally, the cost of grasping should be small indicating that the grasp is successful. Empirically, a threshold (E threshold ) value between 0.5 and 0.8 gives a good result for the implemented cost function. If the distance of the touched object to the palm is less than palmThreshold and the movement of the object due to finger contact is towards the palm, then the palm is used as a virtual finger to counteract the force exerted by the fingers. The negReinforcement parameter specifies the level of punishment returned when a grasp attempt fails (rs neg ). Empirically values greater than −0.1 and less than 0 result in good learning. Generally, a large negative reinforcement overwhelms the positively reinforced plans before they have chance to get represented in the layers.

$$\eqalign{ & {\rm costThreshold} \quad\quad\quad\quad {\rm 0}{\rm .8 } \cr & {\rm palmThreshold } \quad\quad\quad\; {\rm 150 } \cr & {\rm negReinforcement } \quad {\rm - 0}{\rm .05 } \cr} $$
(29)

Exploration and exploitation parameter

α (Randomness) specifies how often to use the learned distribution to generate grasp plans. A value of 1 means always use random parameter selection, while a value of 0 means always generate parameters from the current distribution of the layer. In all the simulations, the Virtual finger layer used the probability distribution representation to generate enclosure parameter (v). Next, we present the parameters used for other layers in the simulation experiments. The default parameters values used as examples in the descriptions above are not repeated here (Tables 

Parameters for SE1a

hand_rotBANK_code_len

N/A

hand_rotPITCH_code_len

N/A

hand_rotHEADING_code_len

N/A

hand_locMER_code_len

10

hand_locPAR_code_len

10

hand_locRAD_code_len

10

MAXREACH

N/A

MAXROTATE

N/A

MAXBABBLE

1000

Reach2Target

INDEX0

costThreshold (E threshold )

0.75

palmThreshold

125

negReinforcement (rs neg )

−0.1

Randomness (α)

0.85

a,

Parameters for SE2

hand_rotBANK_code_len

9

hand_rotPITCH_code_len

9

hand_rotHEADING_code_len

1

hand_locMER_code_len

7

hand_locPAR_code_len

7

hand_locRAD_code_len

1

MAXREACH

1

MAXROTATE

7

MAXBABBLE

45000

Reach2Target

INDEX2

costThreshold (E threshold )

0.80

palmThreshold

150

negReinforcement (rs neg )

−0.05

Randomness (α)

1

b,

Parameters for SE1b

hand_rotBANK_code_len

11

hand_rotPITCH_code_len

11

hand_rotHEADING_code_len

6

hand_locMER_code_len

6

hand_locPAR_code_len

6

hand_locRAD_code_len

1

MAXREACH

N/A

MAXROTATE

55

MAXBABBLE

10000

Reach2Target

INDEX0

costThreshold (E threshold )

0.75

palmThreshold

125

negReinforcement (rs neg )

−0.1

Randomness (α)

0.95

c,

Parameters for SE3

hand_rotBANK_code_len

12

hand_rotPITCH_code_len

7

hand_rotHEADING_code_len

1

hand_locMER_code_len

5

hand_locPAR_code_len

5

hand_locRAD_code_len

5

MAXREACH

1

MAXROTATE

25

MAXBABBLE

20000

Reach2Target

MIDDLE0

costThreshold (E threshold )

0.85

palmThreshold

150

negReinforcement (rs neg )

−0.1

Randomness (α)

0.95

d and

Parameters for SE4

hand_rotBANK_code_len

10

hand_rotPITCH_code_len

7

hand_rotHEADING_code_len

5

hand_locMER_code_len

10

hand_locPAR_code_len

10

hand_locRAD_code_len

1

MAXREACH

10

MAXROTATE

30

MAXBABBLE

10000

Reach2Target

MIDDLE0

costThreshold (E threshold )

0.75

palmThreshold

128

negReinforcement (rs neg )

−0.1

Randomness (α)

1.0

e).

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Oztop, E., Bradley, N.S. & Arbib, M.A. Infant grasp learning: a computational model. Exp Brain Res 158, 480–503 (2004). https://doi.org/10.1007/s00221-004-1914-1

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