INFORMATION

SCIENCES

AN ~I~/RNAT~O~AL JOURNAL

ELSEVIER Information Sciences 116 (1999) 147-164

Inverse kinematics in robotics using neural

networks

Sreenivas Tejomurtula a,1, Subhash Kak b,.

a Avant! Corporation, 46871 BaysMe Parkway, Fremont, CA 94538, USA

b Department of Electrical and Computer EnghleerhTg, Louisiana State University,

Baton Rouge, LA 70803-5901, USA

Received 1 March 1998; accepted 23 October 1998

Communicated by George Georgiou

Abstract

The inverse kinematics problem in robotics requires the determination of the joint

angles for a desired position of the end-effector. For this underconstrained and ill-con-

ditioned problem we propose a solution based on structured neural networks that can be

trained quickly. The proposed method yields multiple and precise solutions and it is

suitable for real-time applications. © 1999 Elsevier Science Inc. All rights reserved.

I. Int roduct i on

Modern robot manipulators, and kinematic mechanisms in general, are

typically constructed by connecting different joints together using rigid links. A

number of links are attached serially by a set of actuated joints. The kinematics

of a robot mani pul at or describes the relationship between the mot i on of the

joints of a mani pul at or and resulting motion of the rigid bodies that form the

robot. Most of the modern manipulators consist of a set of rigid links con-

nected together by a set of joints. Although any type of joint mechanism can be

used to connect the links of a robot, traditionally the joints are chosen from

revolute, prismatic, helical, cylindrical, spherical and planar joints. This paper

looks at manipulators with revolute and prismatic joints.

* Corresponding author. E-mail: kak@ee.lsu.edu

t E-mail: sreeni@avanticorp.com

0020-0255/99/$20.00 © 1999 Elsevier Science Inc. All rights reserved.

PII: S0020- 0255( 98) 10098- 1

148

S. Tejomurtula, S, Kak / hlformation Sciences 116 (1999) 147-164

The different techniques used for solving inverse kinematics can be clas-

sified as algebraic [6,17,14,4,12,18], geometric [10,1,7] and iterative [8]. The

algebraic methods do not guarantee closed form solutions. In case of geo-

metric methods, closed form solutions for the first three joints of the ma-

nipulator must exist geometrically. The iterative methods converge to only a

single solution and this solution depends on the starting point. The most

common neural networks used to solve the problem of inverse kinematics are

error-backpropagation and Kohonen networks. The error-backpropagation

algorithm takes a very long time for forward training. We have proposed a

variant of the error-backpropagation algorithm to solve this problem. This

new approach has the advantage of accuracy over the error-backpropagation

algorithm.

2. Background and notation

The forward kinematics of a robot determines the configuration of the end-

effector (the gripper or tool mounted on the end of the robot), given the relative

configuration of the robot. This paper is restricted to open-chain manipulators

in which the links form a single serial chain and each pair of links is connected

either by a revolute joint or a prismatic (sliding) joint.

The joint space of a manipulator consists of all possible values of the joint

variables of the robot. Specifying the joint angles specifies the location of all

the links of the robot. For revolute joints, the joint variables are given by an

angle q E [a, b) where a and b are angles in radians.

All joint angles are measured using a left-handed coordinate system, so that

angle about a directed axis is positive if it represents an anti-clockwise rotation

as viewed along the direction of the axis. Prismatic joints are described by a

linear displacement along a directed axis.

The number of degrees of freedom of an open-chain manipulator is equal to

the number of joints in the manipulator. For simplicity, all joint variables are

referred to as angles, although both angles and displacements are allowed,

depending on the type of joint. Given a set of joint angles, the determination of

the configuration of the end-effector relative to the base is called forward ki-

nematics.

The workspace of a manipulator is defined as the set of all end-effector

configurations that can be reached by some choice of joint angles. The work-

space is used when planning a task for a manipulator to execute; all desired

motions of the manipulator must remain within the workspace. In this paper,

the range of the possible angles is fixed in advance and the reachable workspace

is calculated.

Given the desired end-effector position, the problem of finding the values of

the joint variables in order for the manipulator to reach that position is inverse

S. Tejomurtula, S. Kak I Information Sciences 116 (1999) 147-164 149

kinematics. This problem may have multiple solutions, a unique solution or no

solution.

2.1. A planar exampl e

To illustrate some of the issues in inverse kinematics, consider the inverse

kinematics of the planar two-link manipulator shown in Fig. 1.

The forward kinematics can be determined using plane geometry.

p, = L, cos(q,) + L2cos(q, + q2), (1)

P2 = L1 sin (ql) + Lzsin(ql + q2). (2)

The inverse problem is to solve for joint variables ql and q2, given the end-

effector coordinates pl and Pz.

q2 = n rk a,

a = cos-' ((L~ + L] - r2)/(2LIL2)),

ql = at an(p2,pl ) 4- b,

b = cos-' ((r 2 + L~ - L~)/2L,r).

(3)

(4)

(5)

(6)

L /

ss s~

Fig. 1. Inverse kinematics of a two-link manipulator.

150 s. Tejomurtula, S. Kak I hformation Sciences 116 (1999) 147-164

2.2. Different methods used f or solv#zg inverse kinematics

The three main methods for solving inverse kinematics, namely, algebraic,

geometric and iterative are described below.

Algebraic: Detailed steps toward an algebraic solution to the PUMA 500

manipulator can be found in Refs. [17,14,4]. To solve for inverse kinematics

algebraically, it is necessary to solve equations q~,q2,..., qu for N degrees of

freedom. The problem can be formulated as follows: given the end-effector

position

= 0 0- l

(7)

where the right-hand side describes the required position and orientation of the

end-effector. The problem comes down to solving N equations for N un-

knowns [14]. This method does not guarantee a closed form solution for a

given manipulator. Thus, engineers usually design simple manipulators where

closed-form solutions exist.Craig [4], Manocha [12] and Zhu [18] proposed a

generalized closed-form solution which can be derived for 6 (or less) DOF

kinematic chain. Manocha [12] outlined a method for solving IK algebraically

using symbolic manipulation to derive univariate polynomial and matrix

computations.

Geometric: As opposed to the algebraic method, a closed form solution

using the geometry of the manipulator is derived. Lee [10] used theorems in

coordinate geometry which can be found in Ref. [1] to derive closed form

solutions for a six DOF manipulator. This involves projecting of the link co-

ordinate frame on the X~_l and Y~_l frame. This method can be applied to any

manipulator with known geometry. The limitation of this method is that the

closed-form solution for the first three joints of the manipulator must exist

geometrically [7]. Apart from that, the closed-form solution for one class of

manipulators cannot be used in other manipulators of a different geometry.

Iterative: This method solves inverse kinematics by iteratively solving for the

joint angles. This method converges to only one solution as opposed to the two

methods presented by Korein and Balder [8]. There are three components that

constitute iterative methods, namely, the Jacobian, pseudo-inverse and mini-

mization methods.

The inverse kinematics problem using neural networks comes under the

class of iterative methods. They are however different from the conventional

iterative methods used for solving inverse kinematics. It is important to note

that the computational requirements are independent of the number of degrees

of freedom of the robot arm; instead they are based on the network archi-

tecture.

S. Tejomurtula, S. Kak I hformation Sciences 116 (1999) 147-164

3. Application of neural networks in inverse kinematics

151

In robotics, solving a problem using a programmed approach requires the

development of software to implement the algorithm or set of rules. Frequently

there are situations as in non-linear or complex multivariable systems, where

the set of rules or required algorithms is unknown or too complex to be

accurately modeled. Even if characterizing algorithms are obtained, they

often are too computationally intensive for practical real-time applications.

To circumvent this problem, neural networks are used. Neural networks are

advantageous because they reduce software development, decrease computa-

tional requirements, and allow for information processing capabilities where

algorithms or rules are not known or cannot be derived.

The computational requirements for task and path planning, and path

control, may be very demanding. However, robotic processes may be formu-

lated in terms of optimization or pattern recognition problems so that neural

network can be adapted.

3.1. Backpropagation

The conventional back-propagation algorithm is as follows: The number of

hidden layers and the number of hidden neurons in each layer are decided. The

network is fully connected between every two adjacent layers, i.e., the input

neurons and the hidden neurons in the first hidden layer are fully connected.

There is a connection between every pair of neurons between hidden layer 1

and hidden layer 2 and so on. The neurons in the final hidden layer are

completely connected to the output layer. The weights of these connections are

chosen at random. The first pattern is fed and its response is transported to the

output layer. The error between the desired output and the actual output is

propagated back. The weights are adjusted iteratively till the error falls below a

threshold. Similarly all input patterns of the training set are fed and weights are

adjusted. This process goes on till all the patterns are stored simultaneously.

The amount of time taken for training makes it practically useless for real

time applications if the training set is very large.

3.2. Neural network inversions

The inversion problem for neural networks is to find inputs that yield a

desired output [11]. There are three commonly used approaches for inverting

networks. These are error back-propogation approach, the optimization ap-

proach and the iterative approach based on update of input vector.

Optimization: In the optimization approach [9], the inversion problem is

formulated as a non-linear programming problem. The neural network is

trained using data points. Once the training is done, the weights are fixed. The

152 S. Tejomurtula, S. Kak / Information Sciences 116 (1999) 147-164

relation between every two hidden layers is approximated as a non-linear

function. These equations are solved along with the constraint inequalities on

the joint variables. A non-linear programming problem where the objective

and the constraint functions can be expressed as a sum of functions, each in-

volving only one variable, is called a non-linear separable programming

problem. It can be approximated as a pseudo-linear programming problem and

solved by a variation of the simplex method, a common technique for solving

programming problems.

lterative: The neural network is trained using given data. Once the training

is done, the weights are fixed. This method is based on the iterative update of

an input node toward a solution, while escaping from local minima. The

update rule is allowed to detect an input vector approaching a local mini-

mum through a phenomenon called update explosion. At or near local

minima, the input vector is guided by an escaping trajectory generated based

on global information, which is predefined or known information on forward

mapping.

Error Back-propagat i on: This algorithm works by adjusting the weights

along the negative of the gradient in weight space of a standard error measure.

The standard error can be the least-mean-square-error of the output. Using

what is essentially the same back-propagation scheme, one may instead com-

pute the gradient of this error measure in the space of input activation vectors;

this gives rise to an algorithm for inverting the mapping performed by a net-

work with specified weights. In this case the error is propagated back to the

input units and it is the activation of these units - rather than the values of the

weights in the network - that are adjusted so that a specified output pattern is

evoked. The inversion is not unique for given targets and depends on the

starting point in input space. The inversion tries to find an input pattern that

generates a specific output pattern with the existing connections. To find the

input, the deviation of each output from the desired output is computed as the

error 6. The error value is used to approach the target input in input space step

by step. The direction and length of this movement are computed by the in-

version algorithm.

The most commonly used error value is the Least Mean Square Error. E LMs

is defined as

9

E LMs = Tp - f wijopi (8)

p=l

The goal of the algorithm, therefore, is to minimize E LMs . Since the error

signal 6pl can be computed as

6pi : Opi(l -- Opi) Z 6pkWik (9)

kESucc(i)

and for the adaption value of the unit activation follows

S. Tejomurtula, S. Kak / Information Sciences 116 (1999) 147-164 153

A netpi = rl6pi resp.netpi = netp/+ rl6pi. (10)

In this implementation, a uniform pattern is applied to the input units in the

first step, whose activation level depends upon the variable input pattern. This

pattern is propagated through the net and generates the initial output O (°). The

difference between the output vector and the target output vector is propagated

backwards through the net as error signal 61(0). This is analogous to propa-

gation of error signals in backpropagation training, with the difference that no

weights are adjusted here. When the error signals reach the input layer, they

represent a gradient in the input space, which gives the direction for the gra-

dient descent. Thereby, the new input vector can be computed as

i(1) = i(o) + ~/. 6~(o), (11)

where q is the step size in the input space. This procedure is now repeated with

the new input vector until the distance between the generated output vector

and the desired output vector falls below the predefined limit of 6m~x, when the

algorithm is halted.

4. Solving inverse kinematics with backpropagation

The inversion with the conventional error-backpropagation algorithm is not

unique for given targets and depends on the starting point in input space.

Lu and Ito [2] proposed the use of subnetworks to obtain more than one

solution for a particular end-effector position. The configuration space was

divided into N regions in a uniform or non-uniform grid. The data points were

generated corresponding to each of the modular configuration spaces.

Fig. 2 shows a planar robot with three degrees of freedom [2] where the

workspace (locus of the end-effector) is in a single plane. The angle q3 is

marked negative because all the angles measured anticlockwise are positive.

The lengths of the arms taken for simulation purposes are:

Ll =0.3, L2=0.25 L3=0.15.

The joint variables are fixed to be in the following ranges.

ql E [-z~/6, 2~/3], q2 E [0, 5r~/6], q3 E I-re/6, ~/6].

The forward kinematic equations of the model are as follows:

pt = LI cos(ql) + L2cos(ql + q2) + L3cos(ql + q2 + q3), (12)

P2 = L, sin (q~) + L2 sin(qt + q2) +L3 sin (q~ + q2 + q3). (13)

The configuration space is divided into eight overlapping regions via the grid

points (-~/6, 3rc/12, 2rc/3), (0, 5~/12, 5~/6) and (-~/6, 0, ~/6). For example,

the first region is described by intervals [re/6, 3rc/12], [0, 5zc/12], and I-n/6,0].

Table 1 shows the ranges of the coordinates of the workspace T~, T2,..., Ts.

154

s. Tejomurtula, S. Kak I hTformation Sciences 116 (1999) 147-1~4

Fig. 2. A three-joint planar arm.

A neural network is trained with the backpropagat i on algorithm to learn the

forward mapping for each of the modul ar configuration spaces. The bound-

aries of the workspace coordinates corresponding to each of the configuration

spaces is determined.

The end-effector position is given as the input. The modul ar networks (T~s)

with which the end-effector position can be reached are identified. The error-

backpropagat i on algorithm is applied to each of those modul ar networks.

The initial guess for the error-backpropagat i on is very critical for a speedy

convergence of the algorithm. Also the initial guess decides to which solution

the algorithm converges. A convenient way to obtain the initial guess is to train

a network with the end-effector positions as the inputs and joint variables as

the output. Corner classification was applied for the calculation of the initial

guess as it is much faster than the back-propagat i on algorithm.

A serious limitation of the back-propagat i on algorithm is the time involved

in training the network to learn the forward kinematics. The number of data

Table 1

An example of the ranges of the outputs in T~

T~ T_, 7"3 T4 7"5 7"6 /'7 T8

p]"~" 0.01 -0.55 -0.20 -0.56 0.06 -0.57 -0.19 -0.58

p~nax 0.70 0.53 0.61 0.18 0.70 0.53 0.59 0.02

p~i, -0.40 0.16 0.13 -0.16 -0.35 0,11 0.04 -0.17

pT "x 0.60 0.70 0.57 0.58 0.60 0.69 0.52 0.59

S. Tejomurtula, S. Kak / hfolvnation Sciences 116 (1999) 147-164

155

points for training cannot be reduced as accuracy is very critical in this ap-

plication. Also forward kinematics can be determined for most of the existing

models of the manipulators. The forward kinematics cannot be determined for

some of the models having redundant joints. Even if the backpropagation

network learns all the data points fed, it cannot beat the accuracy of the for-

ward kinematic equation.

5. A new network architecture for error-backpropagation

Now we present our method which takes advantage of the fact that forward

kinematics can be determined for most of the manipultors.

Any conventional network for backpropagation has weights that are real

numbers. This network deviates from that common rule. The dimensions of the

workspace are trigonometric functions of the joint variables. So some of the

weights of the network are non-linear relationships of the nodes between which

they are connected rather that real numbers. If there is a connection from node

i in hidden layer 1 to node j in hidden layer 2 and the weight is say cosine, the

value at node j in hidden layer 2 is cosine of the value at node i in hidden layer

1. Consider the example of the manipulator shown in Fig. 2. The position of

the end-effector is given by the following set of equations:

Pl = Ll cos (ql) + L2 cos (ql + q2) + L3 cOS (ql + q2 + q3),

(14)

P2 = Ll sin (ql) + Lz sin(ql + q2) + L3 sin(ql + q2 -k- q3),

(15)

where ql, q2 and q3 are the joint variables, p~ and p2 are the coordinates of the

end-effector, Lt,L2 and L3 are the lengths of the robot arms.

Instead of generating data of the joint variables and the Cartesian coordi-

nates for training the network, the variables in the R.H.S of the equation i.e.,

L1, L2, L3, cos, sin are taken as the weights. The network architecture is as

shown in Fig. 3.

Unlike the conventional back-propagation algorithm, no training is re-

quired for this network. This network is different in the sense that some of the

weights are non-linear functions of the nodes between which they are con-

nected. So the error in the output layer cannot be propagated back the usual

way.

5.1. The modified backpropagation

The neural network inversion for the error-backpropagation algorithm

works as follows:

156

S. Tejomurtula, S. Kak / Information Sciences 116 (1999) 147-1~4

HIDDEN HIDDEN

LAYER LAYER

I 2

qC)

k._J ~.~.~ ~ L 3

Fig. 3. Neural network representation of a three-joint planar arm.

hTput: The target point in the work space, range of each of the joint vari-

ables.

Output: The final set of joint variables, within the proper ranges that make

the manipulator reach the target point.

A guess of the joint variables is made. The Cartesian coordinates of the end-

effector position corresponding to the guessed point are calculated. The error in

each of the output neurons is determined. The error is propagated back the

usual way if the weight is of linear nature. If there is a non-linear weight, a

decision is made whether the non-linear function is an increasing or decreasing

function in the neighbourhood being considered. The error is propagated back

with the change of sign if the non-linear function is decreasing. The error is

thus propagated back to the input layer and the input is adjusted. A check is

made whether the joint variables calculated are within the ranges set. If any

joint variable goes beyond its range, the value is clipped to the maximum or

minimum depending on the direction the deviation took place. The forward

calculation is performed. The process is repeated until the RMS value of the

error falls below a threshold. The final updated value of the input joint angles is

S. Tejomurtula, S. Kak I Information Sciences 116 (1999) 147-164

157

the desired result. The algorithm limits the final result to the subrange in which

the guess is made. This algorithm does not require any extensive forward

training as a backpropagation algorithm. It also does not have any encoding

necessary as it takes in real values.

5.2. Modular networks

The partitioning of the joint space is done to get more than one solution, the

forward kinematics of the model involve cosine and sine of the angles ql, ql +

q2 and ql + q2 + q3. Since both the functions are decreasing or increasing de-

pending on the value of the angle, the way the error needs be propagated from

hidden layer 2 to hidden layer 1 (Fig. 3) depends on whether the cosine/sine is

increasing or decreasing. The cosine or the sine of any angle changes sign at

every integer multiple of n/2. So within the ranges of the angles qt, ql + q2, and

ql + q2 + q3, the ranges are divided into smaller intervals with boundaries at

the multiples of n/2. The point to which the algorithm converges depends on

the initial guess. In order to evaluate the solutions for an end-effector position

within the range of the angles, a guess is made for each of the increasing/de-

creasing trend regions. The set of guesses is calculated as follows. In the model

being considered, ql E [-n/6,2n/3]. There are two multiples of n/2 in the

range. So the range is subdivided into [ - n/6, 0], [0, n/2], [2n/3]. Similarly, the

ranges ofql + q2 is divided into [-n/6, 0], [0, n/2], [n/2, n], [n, 3n/2]. The ranges

of ql + q2 + q3 are I-n~3, 0], [0, n/2], [n/2, hi, [n, 3n/2], [3n/2, 5n/3]. n guess is

a set of angles for qt, q2, q3. The guesses are made in such a way that all the

angle ranges for qt, q~ + q2, qt + q2 + q3 are covered.

The ranges of the coordinates of the workspace (T~s) are calculated. An end-

effector position is the input. All the subnets T~ within which the point lies are

identified. It means that there may be zero, one or more solutions in this range.

The other T~'s are ruled out. There may not be a solution in each of the sub-

ranges.

6. Results with robots of different degrees of freedom and different types of joints

The algorithm was applied on different models of the manipulator. It could

produce the solutions to a good degree of accuracy. The models and some of

the results are as follows. The results for the remaining models can be obtained

from [16].

6.1.. A planar robot with three degrees of freedom

The algorithm was applied to evaluate the inverse kinematic solutions of a

planar robot with three degrees of freedom shown in Fig. 2. The accuracy of

158

s. Tejomurtula, S. Kak / Information Sciences 116 (1999) 147-16~I

the sol ut i on, i.e., the RMS di st ance between the cal cul at ed posi t i on of the

end-effector and the posi t i on of the t arget is of the order of 0.00000001.

Tabl e 2 shows the inverse ki nemat i c sol ut i ons for different end-effector po-

sitions.

6.2. A robot arm with three degrees-of-freedom and three-dimentional workspace

These are the results of a r obot arm with three revol ut e j oi nt s and three-

di ment i onal wor kspace [13]. Fig. 4 shows the t hree-j oi nt arm. The same

length of the arms and the same range of the angles are t aken as in the

previ ous section. The forward ki nemat i c equat i ons for this model are as

follows.

Pl = Lt cos (ql) - Lzcos (q2)sin (ql) - L3 cos (q2 + q3)sin (ql), (16)

P2 --- L2 sin (q2) +L3si n(q2 + q3), (17)

P3 = Lt sin(q1) +L2cos( qz) cos( ql ) + Lacos (q2 + q3)cos(ql ). (18)

Unl i ke the previ ous model, the forward ki nemat i cs of this model involves

terms with product s of t ri gonomet ri c terms. So nodes for mul t i pl i ers are in-

t roduced in the net work in hi dden layers as shown in Fig. 5. Redundant nodes

for mul t i pl i ers are i nt roduced in the third hi dden layer to keep a uni form

connect i vi t y bet ween successive layers, i.e., there is no direct connect i on from

Table 2

The inverse kinematic solutions for different end-effector positions for a planar robot

End-effector qt q2 q3 pl p,.

position

1 60.36 51.23 77.76 60.36 60.36

1 -0.523598561 0.389823676 -0.523598776 0.6263140 -0.2750000

1 -0.523598720 0.000000000 0.523598586 0.6263140 -0.2750000

2 0.283428164 2.455173741 -0.012157724 -0.0792008 0.2424363

2 0.283392003 2.450629821 0.000000000 -0.0792008 0.2424363

2 0.349065961 2.617993632 -0.523598776 -0.0792008 0.2424363

2 0.349065822 2.228169666 0.523598776 -0.0792008 0.2424363

3 1.221730545 2.617993792 0.523598776 -0.1402081 -0.0197430

4 1.622054081 1.862576405 -0.239394981 -0.4000000 0.2000000

4 1.612918914 1.780666961 0.000000000 -0.4000000 0.2000000

4 1.656258942 1.938520345 -0.523598776 -0.4000000 0.2000000

4 1.621655383 1.685614178 0.234101156 -0.4000000 0.2000000

4 1.656258947 1.548696341 0.523598776 -0.4000000 0.2000000

5 1.101737701 0.144655985 0.000000000 0.2631111 0.6467348

5 1.178490703 0.083498632 -0.195153053 0.2631111 0.6467348

5 1.126387962 0.049447955 0.139242221 0.2631111 0.6467348

S. Tejomurtula, S. Kak I Information Sciences 116 (1999) 147-164

159

//////////

t

M.,,' q2

I

I

-y

L I

-y

I

L 2 ~rJq3 [-'3

////////

Fig. 4. A three-joint arm.

hidden layer 2 to the output layer. The error is propagated in the usual way

from the output layer to the third hidden layer. At each multiplier node, the

error is propagated back equally on all the incoming links to the node. The

results are as shown in Table 3.

Since the ranges of the joint angles are fixed in advance, the algorithm

evaluates only those solutions that are within the reach with these angle ranges.

Some end-effector positions may have more solutions and some may have

fewer solutions. This is evident from the number of solutions of the first end-

effector position and the second end-effector position.

7. Comparison between the different methods for solving inverse kinematics using

neural networks

The two methods commonly used for manipulator kinematic approximation

are multilayered networks with error backpropagation learning and Kohonen

maps [5]. These approaches are based on computational algorithms that can be

used only after a lengthy and computationally intensive optimization of the

160 S. Tejomurtula, S. Kak I hzformation Sciences 116 (1999) 147-164

HIDDEN HIDDEN

LAYER I I~'~:~R 2

0 a.

HIDDEN

LAYER 9

Fig. 5. Neural network representation of a three-joint arm.

network internal weights (learning). Precision is the criterion considered for

comparison.

Consider an inverse kinematics problem for a three-link anthropomorphic

manipulator. The forward kinematic equations of the manipulator is given by

the following equations:

Pl = (L2 cos (q2) + L3 cos (q2 + q3)) cos (ql),

P2 = (L2 cos (q2) + L3 cos (q2 + q3)) sin (ql),

P3 = L1 + L2 sin(q2) + L3 sin(q2 + q3).

(19)

(20)

(21)

7.1. Multilayered perceptron network

In this method, the network nodes are organized in several layers. The most

common network design is with one or two hidden layers of nodes placed

between input and output layers. Sontag [15] proved that at least two hidden

layers are generally needed for inverse function approximation. The results

being shown are with two hidden layers. The goal is to minimize the mean

square approximation error between the predicted and actual points. The

S. Tejomurtula, S. Kak I Information Sciences 116 (1999) 147-164 161

Table 3

The inverse kinematic solutions for different end-effector positions for a 3-DOF robot and 3-D

!

workspace

End-effector ql q2 q3 Pl P2_ P3

position

1 -0.523598776 0.064951571 -0.523598776 0.46 -0.05 0.19

2 -0.523598776 0.766295849 -0.523598776 0.43 0.21 0.14

2 -0.523598776 0.708162701 -0.375521399 0.43 0,21 0.14

2 -0.523598776 0.388518064 0.488190808 0.43 0.21 0.14

2 -0.523598776 0.564237030 0.000000000 0.43 0.21 0.14

2 -0.523598776 0.467236633 0.263895837 0.43 0.21 0.14

3 -0.523598776 1.530483942 -0.384101928 0.30 0.38 -0.09

3 -0.523598776 1.358596344 0.000000000 0.30 0.38 -0.09

4 -0.008896367 0.279467795 -0.272541291 0.30 0.07 0.38

4 -0.000543558 0.377127807 -0.523598776 0.30 0.07 0.38

4 -0.011870896 0.175905768 0.000000000 0.30 0.07 0.38

4 -0.007871382 0.245411662 -0.183631824 0.30 0.07 0.38

4 -0.001925112 0.000000000 0.492180134 0.30 0.07 0.38

4 -0.004724206 0.058025595 0.349997378 0.30 0.07 0.38

5 -0.521204783 1.557290929 -0.522638798 0.30 0.38 -0.08

5 -0.521204783 !.369091525 0.000924961 0.30 0 -0.08

limitation with this approach is that for more neurons the learning process

does not converge in a reasonable time. The experiments were conducted with

two hidden layers. There were 15 neurons in each of the hidden layers (see

Fig. 6).

7.2. Kohonen maps

Kohonen's self-organizing mapping algorithm is one of the most popular in

robotics application [3]. The Kohonen Map network consists of L nodes

(neurons) with a vector z~ E R k, a vector tp~ E R M, and a gradient matrix W~ E

R M×K associated with each node r. Suppose that a vector x is input to the

network. The node s such that [i x- zsi[ is minimal over all the nodes and ap-

proximates the function

y, = it', + W,(x - z,) (22)

is calculated. The above approximation can be computed once the network

parameters zr, ~,., W~ are known. They can be determined by seeking to mini-

mize the mean square approximation error for the training set. Let the demand

function be

N

D({zr}, {tpr}, { W~}) = (1/2)~-']ly (') - p(U) ll ~ min, (23)

,u=l

162

S. Tejomurtula, S. Kak / Information Sciences 116 (1999) 147-164

MLP: Multilayered Perceptron Network

KM: Kohonen Maps

I0

K.M

I0 ......................... " ..................

MLP

173v ......................................................................... : .........................................................................

e~

165 ....................................................................................................................................................

-6 Neural Network With Non-linear Weights

10

-7

10

I0 I I0 2 I0

NUMBER OF TRAI NI NG POI NTS

Fig. 6. Test set joint space errors.

where fi(") denotes the network output when the input vector x 0') of the training

set pair {x("),y(")} is given as the input.

To find the optimum network parameters iterative minimization of the cost

function with the steepest descent method is used.

7.3. Neur al network with non-linear weights

The limitations with both the approaches above is the time involved in the

forward training and the degree of error involved in the forward approxima-

tion. Both these are overcome in our approach to the problem. The algorithm

presented in this paper has been applied to the above model.

Ll =0.3, L2=0.25, L3=0.15, qlE[--rc/6,2rc/3],

q2 E [0, 5rc/6], q3 E [--~/6, rc/6].

An accuracy (error in the joint variables) of the order of 0.000001 is ob-

tained for this model (Fig. 6).

S. Tejomurtula, S. Kak / Information Sciences 116 (1999) 147-164 163

Table 4

The inverse kinematic solutions for different end-effector positions of a three-link manipulator

q~ q2 q3 pl P2_ P3

0.785398122 1.309012396 0.104674842 0.062346 0.062346 0.689635

0.785398163 1.805810804 -0.314177173 -0.032776 -0.032776 0.692658

0.785398163 2.356196362 0.523591635 -0.227452 -0.227452 0.515600

1.308984846 1.621695644 -0.418871863 0.010674 0.039834 0.689635

1.308984876 1.309000023 0.418872691 0.010674 0.039834 0.689635

1.570796327 1.544005759 -0.314157775 0.000000 0.056854 0.691276

1.570796327 1.308998440 0.314157757 0.000000 0.056854 0.691276

1.308992696 1.308999632 0.209431202 0.018779 0.070083 0.691276

8. Concluding remarks

In this paper, we considered the problem of inverse kinematics in robotics.

We have presented several methods for finding multiple solutions for a given

end-effector position (as in the example of Table 4). The joint space was di-

vided into uniform grids. The conventional back-propagation algorithm was

used for forward training but this leads to several difficulties related to accu-

racy.

We have devised a variant of the conventional error-backpropagation al-

gorithm that overcomes the disadvantages of backpropagation algorithm like

training time and accuracy. Real data can be fed to the algorithm. The network

does not need any training because network weights can be read off using

forward kinematic equations. It could be used for real-time models as it gen-

erates multiple solutions with a very good accuracy.

The proposed variant of the error-backpropagation algorithm could be used

with other types of joints. The choice of the initial guesses is an important

factor in determining the result. The joint angles are analysed and then the

guesses are decided. Then the algorithm is run for each of the guesses. This

gives results of the joint variables some of which are close to each other while

others are quite different. Any one of them could be taken to reach the desired

end-effector position.

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