A RANDOM MATRIX APPROACH TO MANIPULATOR JACOBIAN
Javad Sovizi ∗ Aliakbar Alamdari Automation, Robotics & Mechatronics Laboratory Venkat N. Krovi Department of Mechanical and Aerospace Engineering Automation, Robotics & Mechatronics Laboratory University at Buffalo Department of Mechanical and Aerospace Engineering Buffalo, NY 14260 University at Buffalo Email:
[email protected] Buffalo, New York, 14260 Email:
[email protected] [email protected]
ABSTRACT Traditional kinematic analysis of manipulators, built upon a deterministic articulated kinematic modeling often proves inadequate to capture uncertainties affecting the performance of the real robotic systems. While a probabilistic framework is necessary to characterize the system response variability, the random variable/vector based approaches are unable to effectively and efficiently characterize the system response uncertainties. Hence in this paper, we propose a random matrix formulation for the Jacobian matrix of a robotic system. It facilitates characterization of the uncertainty model using limited system information in addition to taking into account the structural inter-dependencies and kinematic complexity of the manipulator. The random Jacobian matrix is modeled such that it adopts a symmetric positive definite random perturbation matrix. The maximum entropy principle permits characterization of this perturbation matrix in the form of a Wishart distribution with specific parameters. Comparing to the random variable/vector based schemes, the benefits now include: incorporating the kinematic configuration and complexity in the probabilistic formulation, achieving the uncertainty model using limited system information (mean and dispersion parameter), and realizing a faster simulation process. A case study of a 6R serial manipulator (PUMA 560) is presented to highlight the critical aspects of the process. A Monte Carlo analysis is performed to capture the deviations of distal path from the desired trajectory and the statistical analysis on the realizations of the end effector position and orientation shows how the uncertainty propagates throughout the system. ∗ Address
all correspondence to this author.
INTRODUCTION Motion planning of robotic manipulators entailing either direct or inverse kinematics has traditionally been examined in deterministic settings. However, there are several sources of the uncertainties that can substantially affect the overall performance of the robot. These variations arise from inherently uncertain environment, limited range and resolutions of sensors that are also subjected to disturbing noises (sensing uncertainties), failure in accurate actuation (control uncertainties), modeling inaccuracy (simplifications) and computational limitations [1–3]. Hence, a systematic approach to characterize the variations of the system response is required. Many researchers have incorporated the uncertainty effects in the motion planning of the mobile robotic systems [4–9] that generally deals with global localization of the robot subjected to several workspace and uncertainty constraints. Jung and Ravani [10] considered the uncertainty in Jacobian matrix due to ignoring the actuator Jacobian (mapping from actuator space to joint space). They used neural network method for online compensation of the uncertainty in the Jacobian matrix. Cheah et al. [11] investigated the kinematic as well as dynamic uncertainties in tracking control of the robot end effector (EE). The uncertain kinematic and dynamic parameters of the system are updated based on an online updating law that results in convergence to the desired EE trajectory. Other works as [12–14] have also used the estimation of Jacobian matrix due to kinematic uncertainties. Wang and Chirikjian [15] developed a nonparametric second order approximation of error propagation whose one of its main applications is in the robot kinematic analysis and motion
planning in presence of uncertainties. The main part of the paper discusses efficient approximation of mean and covariance of two uncertain frame density functions convolution, while the mean and covariance of each random frame are known. The propagation of spatial uncertainties covariance using first order approximation method has also been studied in an earlier work by Su and Lee [16]. The main objective of this paper is to develop a systematic way to characterize the propagation of uncertainties in a manipulator system. In this work, we focus on manipulator jointactuation uncertainties, which can be induced by joint flexibilities, joint clearances, imperfections, complex motor assemblies, etc. However, we assume that: (i) at each time step the joint space variables can be accurately observed by the sensors; and (ii), the structure of the manipulator is known and the exact EE position can be obtained corresponding to a given realization of random joint space configuration through forward kinematic relations. So, at each time step (tk ) the joint space variables are known (accurately observed by the sensors) and the configuration of the system at subsequent time step (tk+1 ) is uncertain (due to actuation uncertainties). The novelty of our work arises from formulating the Jacobian matrix of the manipulator as a random matrix. The process noise acting on the uncertain system is modeled by randomness of the kinematic structure Jacobian matrix. This offers several significant advantages when compared to the random variable/vector based approaches. First, the proposed method incorporates the kinematic configuration (and complexity of the manipulator structure) within the probabilistic formulation that facilitates a better characterization of real system uncertainties. In contrast, in the random variable/vector based methods, probabilistic formulation of the system uncertainties does not depend on the structural configuration/complexity of the robot. Second, the random matrix formulation is based on maximum entropy method (developed in next sections) and uses the available system information. In the subsequent sections, we will show that the only information used to construct the probability model of the Jacobian matrix are its mean and a scalar value dispersion parameter that represents the level of uncertainty in the system. The mean can be determined by substituting the joint variables values into closed-form formulation of the Jacobian; while the scalar dispersion can be estimated from system knowledge or from experimental measurements. In addition, this method provides a faster simulation by eliminating the substitution of the realizations of the random joint variables into nonlinear functions. To the best of our knowledge, except the work we developed in [17], this is the first time that random matrix theory is employed to model the kinematic Jacobian matrix of the uncertain manipulator systems. The rest of this paper is organized as follows: In the next section, a basic background in manipulators forward and inverse kinematics is reviewed and numerical method to solve the inverse kinematic problem is presented. The probabilistic formulation of Jacobian matrix using random matrix theory is developed im-
mediately afterwards. Next, PUMA 560 serial manipulator is specifically considered to implement the developed method. A three-dimensional path following task is assumed for the robot EE and the Monte Carlo numerical simulation is performed to obtain sufficient number of system response realizations that are statistically analyzed. Finally, the discussion is presented in the last section.
FORWARD AND INVERSE KINEMATICS A routine problem in manipulators kinematic analysis is to express the end effector position and orientation as a function of joint space variables (forward kinematics) and vice versa (inverse kinematics). The forward kinematic relation may be stated as:
x(t) = g(q(t))
(1)
where x is the vector of position and orientation of the EE, q is vector of joint variables, t is time and g is a vector function that is, in general, nonlinear and maps from joint space to operational space. In forthcoming equations, capital letter is used for matrix notation and bold letters indicate the random vector or matrix. Given the trajectory of the end effector, the inverse of function g, if exists, can be used to obtain the joint variables as a function of time. However, in many cases g−1 does not exist or is not explicitly available and requires high level of algebraic and geometric intuitions. So, numerical methods, that are applicable to all kinematic structures, are exploited. The numerical scheme used in this paper employs the differential kinematic relation described by Eq. (2).
x(t) ˙ = J(q(t))q(t) ˙
(2)
where J is Jacobian matrix of the kinematic structure. From Eq. (2) one can write:
q(t) ˙ = J −1 (q(t))x˙d (t)
(3)
that defines inverse differential kinematic (IDK) relation in which x˙d (t) is desired EE velocity vector. Then, joint variable vector, q(t) can be implicitly expressed as:
Z t
q(t) =
J −1 (q(τ))x˙d (τ)dτ + q(0)
(4)
0
Equation (4) can be numerically solved in a discrete time domain as follows (Euler integration):
q(tk+1 ) = q(tk ) + q(t ˙ k )∆t and k = 0, . . . , m
(5)
where tk+1 = tk + ∆t. Substituting Eq. (3) in Eq. (5) gives:
q(tk+1 ) = q(tk ) + J −1 (q(tk ))x˙d (tk )∆t
(6)
Thus, given the end effector trajectory, xd (t), and the Jacobian of the manipulator, J(q(t)), the joint variables can be determined in discrete time points, tk ’s. The Jacobian matrix in Eq. (6) can be considered as a random matrix at several time points (tk ’s) in which the system is subjected to the uncertainties. The next section establishes a probabilistic formulation of the Jacobian matrix using random matrix theory.
where LA is an upper triangular matrix corresponding to Cholesky decomposition of A, the mean of random matrix A, and GA is a random symmetric positive definite matrix (GA ∈ S M+ n) with identity mean. Now, although the Jacobian matrix is in general rectangular and can also be singular, but in several common cases the number of joint space variables is identical to the number of EE degrees of freedom, i.e. the Jacobian matrix is square. In addition, the task of the manipulator is commonly defined such that the manipulator does not reach the singularity regions. Then, the Jacobian matrix is non-singular during the operation time. So, let for simplicity assume the Jacobian matrix of kinematic structure is non-singular square matrix with dimensions n, J ∈ Mn . In a similar fashion, we decompose the mean of Jacobian matrix as:
J = LJ U J
(9)
RANDOM JACOBIAN MATRICES Dealing with a random system, Eq. (6) can be written as follows:
Equation (9) is LU decomposition of Jacobian mean matrix in which U J is and upper triangular matrix and LJ is such that LJ U J = J. Now, for random Jacobian matrix we can write:
q(tk+1 ) = q(tk ) + J−1 (q(tk ))x˙d (tk )∆t
J = LJ BU J
(7)
In fact, as mentioned in preceding sections, the joint configuration of the system at the current time (q(tk )) is observed accurately by the sensors. However, due to the actuation uncertainties the Jacobian matrix is random (shown by bold letter) and consequently the joint space configuration at subsequent time (q(tk+1 )) is random. Note that this enables a decoupling of uncertainties. The uncertainties arising from choice of numerical integration scheme and time step can be treated independently from the uncertainties introduced via the Jacobian. Now using forward kinematic relation defined by Eq. (1), the realization of the EE random position and orientation can be obtained by substituting the realization of the joint space variables at each time step. To utilize the developed method, one needs to first construct the probabilistic formulation of the random Jacobian matrix. Maximum Entropy Method and Random Jacobian Matrix Formulation Before modeling the random Jacobian matrix, we first refer to a method proposed by Soize [18, 19] to construct the random symmetric positive definite matrices of structural systems. The n × n random symmetric and positive definite matrix A, denoted as A ∈ S M+ n is written as:
(10)
where B ∈ S M+ n . In fact, Eq. (10) states that the perturbation of the Jacobian matrix is symmetric and positive definite although the Jacobian matrix is not (in general). From Eq. (10) the expected value of random matrix B can be obtained as follow:
E[J] = E[LJ BU J ] = LJ E[B]U J
(11)
Equating the right hand side of Eq. (9) and Eq. (11) gives:
E[B] = In
(12)
So, the available information and constraint on the density function of random matrix B are: 1. Z
E [B] =
B∈S M+ n
BpB (B)dB = B = In
(13)
pB (B)dB = 1
(14)
2. Z
A = LTA GA LA
(8)
B∈S M+ n
Equation (14) is called the normalization constraint of pB (B). To find appropriate matrix variate distribution, an information-theoretic approach is employed. In this approach, all information about the matrix as well as its physical constraints are first considered. Then, the density function is obtained using maximum entropy principle proposed by Jaynes [20] in which the entropy measure, initially introduced by Shanonn [21] for discrete random variables, is maximized subjected to the existing constraints. In fact, the probability density function derived through this method provides maximum uncertainty while satisfying the constraints. So, the more constraints the less variations in the density function. Maximum entropy method [22] has been frequently utilized to obtain the density function of the random systems parameters. Soize [18, 19, 23] and Adhikari [24] used the maximum entropy method to derive the probability model of the structural system matrices (mass, damping and stiffness). In this paper, similar method is followed to find the matrix-variate density function of random matrix B. Entropy measure for density function pB (B) is defined as:
S(p) = −
Z B∈S M+ m
pB (B) ln{pB (B)}dB
(15)
Lagrangian multiplier method can be used to maximize S(p) subjected to the constraints in Eq. (13) and (14). The Lagrangian functional is constructed as:
L (pB (B)) = −
Z
pB (B) ln{pB (B)}dB Z pB (B)dB − 1 − (λ0 − 1) B∈S M+ n Z Bp (B)dB − B − tr Λ1 B +
−1 1 1 det (Σ) 2 p ) det (S) 2 (p−n−1) etr − 21 Σ−1 S
where S ∈ S M+ n, p≥n
(18)
Comparing Eq. (17) with Eq. (18) shows that B has a B Wishart distribution with parameters n, p = n + 1 and Σ = n + 1 . The distribution of B derived as above is maximally uncertain [24] as the minimum number of constraints are used to obtain the density function. Determining the final distribution may depend upon the number of added constraints. The example of such constraints that can be applied to the random matrix B is the existence of its inverse moments that implies the existence of the moment and pdf of the response. This constraint is written as:
E[kB−1 kνF ] < ∞
(19)
where k.kF is Frobenius norm and ν is the order of inverse moment. It can be shown [18, 24] that in this case random matrix B turns out to have a Wishart density function with parameters n, B p = θ + n + 1 and Σ = θ + n + 1 where θ = 2ν. So, from the construction of random matrix B density function, only selection of θ remains. To this end, the normalized standard deviation of B is defined as [18, 24]:
σ2B =
E[kB − E[B]k2F ] kE[B]k2F
(20)
(16) 1 θ= 2 σB
{tr (B)}2 1+ tr B2
! (21)
n
where λ0 ∈ R and Λ1 ∈ S M+ n are Lagrange multipliers corresponding to the normalization and mean constraints, respectively. Using the variational and matrix calculus it can be shown that (see [24]):
pB (B) =
1 2p
After some algebra, θ can be derived from Eq. (20) as:
B∈S M+ n
B∈S M
1 pS (S) = 2 2 pn Γn
rnr det (B)−r 1 etr{−rB−1 B} and r = (n + 1) (17) Γn (r) 2
Definition 1: An n × n random symmetric positive definite matrix S is said to have a Wishart distribution with parameters n, p and Σ ∈ S M+ n if its probability density function (pdf) is given by [25]:
σB , called dispersion parameter of random matrix B, is a measure of uncertainty that was initially introduced by Soize [18]. This value can be assumed from experience or experimental measurements. Having the Wishart parameters, samples of random matrix B can be simply generated using MATLAB function wishrnd. Matlab automatically converts the p parameter to the nearest integer for non-integer values of θ obtained from Eq. (21). Then, the samples of the random Jacobian matrix can be obtained by substituting the B samples in Eq. (10).
NUMERICAL STUDY: PUMA 560 INDUSTRIAL ROBOT To implement the proposed method in the forgoing sections, a serial manipulator with 6 revolute joints, PUMA 560 robot,
J J J = vL vU JωL JωU
(22)
where JvL and JvU are block matrices that relate the linear EE velocities to the lower (q1 , q2 and q3 ) and upper (q4 , q5 and q6 ) three joint variables, respectively. Similarly the angular velocities are related to three lower and upper joint variables through blocks JωL and JωU , respectively. In a deterministic system, because the wrist rotations (upper three joint variables) do not contribute in EE position then JvU = 0. However, in random Jacobian matrix, the components of this block are not in general zero. This implies that in the random system due to uncertainties in the manipulator chain, the wrist joints variables may affect the EE position. The friction at the joints, imperfections, etc can origin these sorts of uncertainties. Figure 1: PUMA 560 CONFIGURATION. Table 1: DENAVIT-HARTENBERG PARAMETERS OF PUMA 560.
i
θ
d
a
α
1
q1
0
0
1.571
2
q2
0
0.4318
0
3
q3
0.15
0.0203
−1.571
4
q4
0.4318
0
1.571
5
q5
0
0
−1.571
6
q6
0
0
0
is considered. PUMA 560 is a wrist partitioned manipulator in which the last three revolute joints axes intersect at the wrist center. Obviously, these joints variables do not contribute in the position of EE but in the orientation of EE frame. Its waist, shoulder and elbow rotations along with rotations at the wrist resemble the motion of human arm. Figure 1 shows an industrial PUMA 560 and the joint space coordinate system assignments. The standard Denavit-Hartenberg parameters of PUMA 560 are summerized in Tab. 1. The forward and inverse kinematic equations of PUMA 560 as well as its Jacobian derivation can be found in the literature. The Jacobian matrix can be partitioned as [26]:
Here, a MATLAB robotic toolbox [27] is utilized to model the PUMA 560 and calculate the Jacobian in different time points. The zero joint angle configuration as well as nominal non-singular configuration plotted by MATLAB toolbox are shown in Fig. 2. The joint variables in nominal configuration are q1 = 0, q2 = 45◦ , q3 = 180◦ , q4 = 0, q5 = 45◦ and q6 = 0. Now, for Monte Carlo numerical implementation of the developed method a desired EE trajectory is first defined. A time interval [tl ,tu ] ⊂ [t0 ,tm ] (where t0 and tm are respectively the times corresponding to the start and the end of the operation) is assumed in which the system is disturbed by uncertainty sources. This interval can be determined by having a prior knowledge of the system or from the experiment. Then, for each simulation (realization), for t ∈ [tl + ∆t,tu ] Eq. (7) is solved in which the Jacobian matrix (at each time step) is random and its generated sample is substituted. To generate the sample, the Jacobian mean matrix is calculated by substituting q(tk ) (we assume q(tk ) is accurately sensed by sensors in real system so in the simulation we substitute the obtained realization of q(tk ) from previous time step) in the Jacobian formulation. Then the elements of the LU decomposition of the Jacobian mean matrix as well as a sample of the perturbation matrix, B, generated based on the developed method in preceding section, are used in the Eq. (10). Moreover, for t ∈ [0,tl ] and t ∈ [tu + ∆t,tm ] the joint space variables are calculated through Eq. (6) in which the Jacobian matrix is deterministic. Having the realization of the q(t) for entire simulation time, assuming the structure of the manipulator is known, the realization of random EE position and orientation can be obtained using the forward kinematic relations. The desired EE trajectory is defined as:
a
b
Figure 2: a) ZERO ANGLE CONFIGURATION b) NOMINAL NON-SINGULAR CONFIGURATION
x0EE x0EE (0) −0.02t y0 EE 0.02t y0 EE (0) z0EE −0.02t z0EE (0) and t ∈ [0, 5] xEE (t) = + = Φx0EE 0.05t Φx0EE (0) Φy0EE 0.05t Φy0EE (0) 0.05t Φz0EE (0) Φz0EE (23) At starting time (t = 0), the manipulator has the nominal configuration shown in Fig. (2) b. So, using forward kinematics we have:
x0EE (0) 0.5963 m y0 EE (0) −0.15 m z0EE (0) −0.0144 m = Φx (0) 0EE π 0 Φy0EE (0) 2 rad 0 Φz0EE (0)
Note that we use Euler angles to express the orientation and angular velocities of the EE frame. The uncertainty time interval for the analysis is assumed to be [3, 5] s. In addition, we have chosen ∆t = 0.001 s and σB = 0.6. 2000 realizations of EE position and orientation are obtained using the developed method. Figure (3) and (4) show
the kernel density estimate of the obtained EE position and orientation in the base frame coordinates at t = 3.5 s and t = 5 s, respectively. Note that in all density plots, the dash lines represent the accurate (desired) value of the corresponding variable. Figure (5) shows the EE spatial position in the base frame for all 2000 realizations at t = 3.5 s and t = 5 s. As it can be seen from the Fig. (5), a few EE spatial position realizations are scattered around the region containing most of the realizations. This can also be inferred from the extended tails of the density plots. Occurrence of these points although is with very low probability but can depend on the kinematic structure, level of disturbances (dispersion parameter), desired task of the robot etc.
The desired values, means and standard deviations corresponding to all above cases are summarized in Tab. 2. The third row of the tables corresponding to t = 3.5 s and t = 5 s shows the deviation of mean from desired non-erroneous values. It can be seen that a significant deviation exists in most of the EE position and orientation quantities in both time points. This deviation also changes with different factors as mentioned above (kinematic structure, task, dispersion parameter etc.).
Due to accumulation of uncertainties along the serial chains, parallel-architectures provide less uncertainty propagation comparing to their serial counterparts [17].
350
350
300
300
300
250
250
250
200
200
150
150
100
100
50
50
200 150 100
0 0.5
0.51
0.52 x0
0.53 (m)
0.54
0.55
0 −0.095
50
−0.09
EE
−0.085 y0
−0.08 (m)
−0.075
−0.07
0 −0.095
a
b 350
300
300
300
250
250
250
200
200
200
150
150
150
100
100
100
50
50
50
0.18
0.185 0.19 Φx (rad)
0.195
0.2
0.205
0 1.745
1.75
0
EE
d
1.755 Φy
−0.08 (m)
−0.075
−0.07
c
350
0.175
−0.085 z0
EE
350
0 0.17
−0.09
EE
1.76 (rad)
1.765
1.77
0 0.17
0.18
0.19
0
Φz
0.2 (rad)
0.21
0.22
0.23
0
EE
EE
e
f
Figure 3: KERNEL DENSITY ESTIMATE OF a) x0EE b) y0 EE c) z0EE d) Φx0EE e) Φy0EE f) Φz0EE AT t = 3.5 s.
DISCUSSION In this paper, for the first time (except our work in [17]) the random matrix theory was employed to formulate the Jacobian matrix of random manipulators that provides a probabilistic framework to study the effects of uncertainties in the kinematic systems. This approach 1) allows to consider the kinematic structure and complexity of the robotic system in the probabilistic formulation 2) facilitates the construction of the probabilistic model of the random Jacobian matrix with limited available information (mean and dispersion parameter) 3) and also provides faster Monte Carlo analysis by elimination of solving several nonlinear equations. LU Factorization permits the mean Jacobian matrix (obtained from substituting the joint variables values into the Jacobian formulation) to be decomposed such that a symmetric and positive definite random perturbation matrix with identity mean could be adopted to model the random Jacobian matrix. Next, the maximum entropy principle was utilized to derive the density function of random perturbation matrix given its mean (identity matrix) and the dispersion parameter that represents the level of the uncertainty of the system (process noise). It was shown that the random perturbation matrix turns out to have a Wishart distribution. The parameters of the distribution were explicitly expressed in terms of the mean, its dimension and the dispersion parameter. So, by choosing only a scalar value (dispersion parameter) and having the mean matrix, the random matrix density
function can be readily characterized. Having the probabilistic formulation of random Jacobian matrix, Monte Carlo analysis was performed for a 6R serial manipulator, PUMA 560, to obtain the realizations of the EE position and orientation. The statistical analysis was performed on the response of the system in different fixed choices of simulation time. Moreover, the kernel density estimates were used to show how the uncertainty is reflected in the system response.
FUTURE WORK In this paper we emphasized on utilizing the random matrix formulation that takes into account the kinematic structure and configuration complexity of the robotic system. However, its performance has not yet been quantitatively compared with the similar analyses based on random variable/vector schemes and experimental data.
ACKNOWLEDGMENT This work was partially supported by the National Science Foundation award CNS-1314484.
180
200
140
160
120
140
150
100
120
80
100 100
80
60
60 40
50
40
20
20 0 0.43
0.44
0.45
0.46 x0
0.47 (m)
0.48
0.49
0 −0.07
0.5
−0.06
EE
−0.05 y0
−0.04 (m)
−0.03
0 −0.15
−0.02
−0.14
−0.13
EE
a
−0.11
−0.1
−0.09
EE
b
c
180
160
−0.12 z0 (m)
150
160 140
140 120
120 100
100
100
80
80
60
60
40
40
20
50
20
0 0.24
0.26
0.28 Φx
0.3 (rad)
0.32
0
0.34
1.82
1.84
1.86
Φy
1.88 1.9 (rad)
1.92
0 0.24
1.94
0.26
0.28
0
0
d
0.3 Φz
0.32 (rad)
0.34
0.36
0.38
0
EE
EE
EE
e
f
Figure 4: KERNEL DENSITY ESTIMATE OF a) x0EE b) y0 EE c) z0EE d) Φx0EE e) Φy0EE f) Φz0EE AT t = 5 s.
−0.07
−0.08
−0.075
EE
(m)
−0.1
−0.085
z0
z0
EE
(m)
−0.08
−0.12
−0.09 −0.095 −0.1
0.5 −0.09
0.55
−0.08 y0
(m)
−0.14 −0.08
0.42 −0.06
0.44 −0.04
−0.07 0.6
x0
(m)
EE
EE
0.46 −0.02 0.48
y0
(m)
x0
(m)
EE
EE
a
b
Figure 5: RANDOM SPATIAL POSITION OF EE AT a) t = 3.5 s b) t = 5 s
REFERENCES [1] Erdmann, M. A., 1984. On Motion Planning with Uncertainty. Tech. Rep. AITR-810, Artificial Intelligence Laboratory, MIT, Aug. See also URL http://hdl.handle.net/1721.1/6949. [2] Donald, B. R., 1987. Error Detection and Recovery in Robotics. Springer-Verlag.
[3] Thrun, S., Burgard, W., and Fox, D., 2005. Probabilistic Robotics. MIT Press. [4] Erdmann, M. A., 1985. “Using backprojections for fine motion planning with uncertainty”. In Proceedings of IEEE International Conference on Robotics & Automation, Vol. 1, pp. 549–554. [5] Donald, B. R., 1987. “Error Detection and Re-
Table 2: STATISTICS OF EE POSITION AND ORIENTATION t = 3.5 s x0EE
y0 EE
z0EE
Φx0EE
Φy0EE
Φx0EE
Desired value
0.5263 (m)
-0.08 (m)
-0.0844 (m)
10.0268 (deg)
100.0268 (deg)
10.0268 (deg)
Mean |Mean − Desired value| × 100 |Desired value|
0.5067 (m)
-0.0865 (m)
-0.0768 (m)
11.2012 (deg)
100.9148 (deg)
11.8933 (deg)
3.72%
8.12%
9%
11.71%
0.89%
18.62%
Standard deviation
1.4 (mm)
1.31 (mm)
1.65 (mm)
0.0753 (deg)
0.071 (deg)
0.0934 (deg)
t =5s
[6]
[7]
[8]
[9]
[10]
[11]
[12]
x0EE
y0 EE
z0EE
Φx0EE
Φy0EE
Φx0EE
Desired value
0.4963 (m)
-0.05 (m)
-0.1144 (m)
14.3239 (deg)
104.3239 (deg)
14.3239 (deg)
Mean |Mean − Desired value| × 100 |Desired value|
0.4408 (m)
-0.0612 (m)
-0.1009 (m)
18.6202 (deg)
107.6978 (deg)
20.466 (deg)
11.18%
22.4%
11.8%
29.99%
3.23%
42.88%
Standard deviation
2.87 (mm)
2.59 (mm)
3.31 (mm)
0.1608 (deg)
0.1371 (deg)
0.2359 (deg)
covery for Robot Motion Planning with Uncertainty”. PhD Thesis, Massachusetts Institute of Technology, Cambridge, MA, Jul. See also URL http://hdl.handle.net/1721.1/6851. Smith, R., and Cheeseman, P., 1986. “On the Representation of and Estimation of Spatial Uncertainty”. International Journal of Robotics Research, 5, pp. 56–68. Fraichard, T., and Mermond, R., 1998. “Path planning with uncertainty for car-like robots”. In Proceedings of IEEE International Conference on Robotics & Automation, Vol. 1, pp. 27–32. Miura, J., and Shirai, Y., 2000. “Modeling motion uncertainty of moving obstacles for robot motion planning”. In Proceedings of IEEE International Conference on Robotics & Automation, Vol. 3, pp. 2258–2263. Kurniawati, H., Bandyopadhyay, T., and Patrikalakis, N. M., 2012. “Global Motion Planning Under Uncertain Motion, Sensing, and Environment Map”. Autonomous Robots, 33(3), Jun., pp. 255–272. Jung, S., and Ravani, B., 1998. “Online kinematic jacobian uncertainty compensation for robot manipulators using neural network”. In Proceedings of IEEE International Conference on Systems, Man, and Cybernetics, Vol. 4, pp. 3483–3488. Cheah, C. C., Liu, C., and Slotine, J. J. E., 2006. “Adaptive Tracking Control for Robots with Unknown Kinematic and Dynamic Properties”. International Journal of Robotics Research, 25(3), Mar., pp. 283–296. Doulgeri, Z., and Arimoto, S., 1999. “A force control for a
[13]
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