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Estimation of robot states with poisson process based on EKF approximate of Kushner filter: a completely coordinate free Lie group approach

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Abstract

In this paper, a Lie coordinate-free torque based Euler–Lagrange equations of motion are developed for a 3-D link (3-DOF) robot. Intentional torque and jerky torque (non-intentional torque) are considered as the inputs to the dynamic profile of the robot. The jerky torque is modelled as a superposition of compound Poisson processes, which is a unique feature. The state vector of the robot, i.e., angular position and angular velocity vector, is thus a Markov process whose transition probability generator can be expressed in terms of the rate of the compound Poisson process that defines the jerky torque. Proof of frame invariance is provided to support the coordinate-free robot dynamics profile. Noise-free measurement is investigated as an ideal case. Angular position measurement is considered with white Gaussian noise. Further, an implementable finite-dimensional EKF approximate to Kushner–Kallianpur filter is obtained to estimate the robot state vector. Finally, the simulations are implemented on commercially available Omni bundle robot.

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Abbreviations

R(t):

Rotation matrix after time t

SO(3):

The group of proper rotations of 3-D space

V(t, r):

Velocity function

K(t):

Link kinetic energy

J :

Moment of inertia tensor, which is constant

\(\rho\) :

Density of 3-D link

\(N_k(t)\) :

Poisson process

\(\begin{bmatrix} \alpha _k \\ \beta _k \\ \gamma _k \end{bmatrix}\) :

Strength of kth Poisson jerk torque

\(\tau _\phi , \tau _\theta ,\tau _\psi\) :

Torque components

\({\hat{e}}_3\) :

Unit vector along ‘z’ direction

\({\mathcal {L}}\) :

Lagrangian

[X, Y]:

Commutation of matrices X, Y

\(X_1,X_2,X_3\) :

Lie group generator of SO(3) group

\(\phi (t),\theta (t),\psi (t)\) :

Angular positions

\(U(\phi ,\theta ,\psi )\) :

Gravitational potential energy

\(\Delta\) :

Time discretization interval

W(t):

White compound poisson noise process

\({\mathbb {E}}\) :

Expectation operator

\(\lambda _k\) :

Rate of Poisson process \(N_k(t)\)

\(P_{\delta W}\) :

PDF of compound Poisson noise differential

ml, ML :

Maximum likelihood

\(\omega (t)\) :

Angular velocity

\({\hat{q}}_t\) :

Estimate of q(t) given measurements upto time t

\({\hat{V}}_t\) :

Estimate of V(t) given measurements upto time t

Cov(X):

Covariance matrix of random vector X

\(\pi _t\) :

Conditional expectation of measurements upto time t

\(Z_t\) :

Measurements upto time t

dz(t):

Measurements differential

K :

Infinitesimal generator of the Markov process \(\begin{bmatrix} q(t)\\ \omega (t) \end{bmatrix}\)

\(dv_t\) :

Differential of measurements noise

\(h_t(q,V)\) :

Measurements observable at time t in the absence of measurement noise

\(H_t\) :

Jacobian matrix of \(h_t\) equivalent to state estimation \(({\hat{q}}_t,{\hat{V}}_t)\)

P(t):

State estimation error covariance matrix

F(q, V):

Driving force/drift of (q, V)

G(q):

Poisson jerk noise coefficients matrix

ad(X):

Adjoint operator of the Lie-algebra element X

\(\epsilon (kmr)\) :

Completely antisymmetric symbol. \(\epsilon (abc) = 0\) if any of the two indices of a, b, c are equal

\(C_{km}\) :

Coefficients of the quadratic form in the kinetic energy matrix relative to modified angular velocities \(\alpha ,\beta ,\gamma\)

\(M_{ab}\) :

Mass moment of kinetic matrices relative to angular velocities \({\dot{\phi }},{\dot{\theta }},{\dot{\psi }}\)

\({\mathbb {T}}\) :

Anti-symmetric torque tensor/matrix

\(\tilde{{\mathbb {T}}}\) :

Time integral of \({\mathbb {T}}\)

\(\Omega (t)\) :

Anti-symmetric angular velocity tensor/matrix

WGN:

White Gaussian noise

WCPN:

White Compound Poisson noise

CRLB:

Cramer–Rao lower bound

MLE:

Maximum likelihood estimation

PDF:

Probability density function

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Authors and Affiliations

  1. Instrumentation and Control Engineering Department, Netaji Subhas University of Technology (Formerly Netaji Subhas Institute of Technology), New Delhi, India

    Rohit Rana & Prerna Gaur

  2. Mechanical Engineering Department, Netaji Subhas University of Technology (Formerly Netaji Subhas Institute of Technology), New Delhi, India

    Vijyant Agarwal

  3. Electronics and Communication Engineering Department, Netaji Subhas University of Technology (Formerly Netaji Subhas Institute of Technology), New Delhi, India

    Harish Parthasarathy

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  1. Rohit Rana
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Rana, R., Gaur, P., Agarwal, V. et al. Estimation of robot states with poisson process based on EKF approximate of Kushner filter: a completely coordinate free Lie group approach. Meccanica 56, 1239–1261 (2021). https://doi.org/10.1007/s11012-021-01325-3

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