Extended and Unscented Kalman Filtering Applied to a Flexible-Joint Robot with Jerk Estimation

Abstract

Robust nonlinear control of flexible-joint robots requires that the link position, velocity, acceleration, and jerk be available. In this paper, we derive the dynamic model of a nonlinear flexible-joint robot based on the governing Euler-Lagrange equations and propose extended and unscented Kalman filters to estimate the link acceleration and jerk from position and velocity measurements. Both observers are designed for the same model and run with the same covariance matrices under the same initial conditions. A five-bar linkage robot with revolute flexible joints is considered as a case study. Simulation results verify the effectiveness of the proposed filters.

1. Introduction

In recent years, many researchers have investigated the control problem of robots with flexible joints. However, compared to the large volume of the literature available on control of rigid robots, relatively little has been published on the control of flexible-joint robots. On the other hand, for a robot manipulator to carry out demanding tasks with high performance, such as the space robots performing services in space, joint flexibility due to gear elasticity, shaft windup and use of harmonic drives, has to be taken into account in both modeling and control of robot manipulators. As the experimental investigations by Sweet and Good [1] show, the effects of joint flexibility can limit the robustness and performance of a given robot controller and can even lead to instability if neglected in the controller design. Moreover, the joint flexibility can serve as the first approximation of robot link flexibility [2] and hence the study of joint flexibility can provide another perspective into the study of flexible link which allows much lighter link weight and thus much faster motion of the robot. If we assume that the flexibility is modeled as a linear torsional spring, we obtain the dynamic model of the manipulator with flexible joints. Flexible robotic manipulators have several advantages over rigid manipulators depending on specific applications; some of these advantages are as follows: low actuator drive requirement, high speeds, low weight, small size, and low cost [3–6].

The robust nonlinear control of these robots based on the state feedback requires the knowledge of state variables for each joint, which may be either positions or velocities of the motors and of the links or positions, velocities, accelerations and jerks of the links [5]. Not only is the full state measurement costly, but there are problems with link acceleration and jerk. Since the former is difficult and the latter impossible to measure, we are forced to consider an observer which provides estimations for these unmeasurable states from position and velocity measurements [7].

Within the significant toolbox of mathematical tools that can be used for stochastic estimation from noisy sensor measurements, one of the most well-known and often used tools is the Kalman filter. As an extension to the same idea, the extended Kalman filter (EKF) is used if the dynamic of the system and/or the output dynamic is nonlinear. It is based on linearization about the current estimation error mean and covariance [8]. Although it is straightforward and simple, it has drawbacks too [9]. The Unscented Kalman Filter (UKF) is the newest revision of the Kalman Filter, proposed to overcome these flaws. It does not need the linearization for a nonlinear function and is more accurate and simpler than the EKF applied to nonlinear systems [10, 11].

In this paper, we derive the dynamic model of a five-bar linkage robot with flexible joints and propose a state-space model for the robot. Then, we apply extended and unscented Kalman filters to estimate the proposed state for the robot. The augmented state is herein composed of the position, velocity, acceleration and jerk of the links. Computer simulations are well performed to verify the performances of the proposed filters.

The remainder of this paper is organized as follows. Section 2 presents the dynamic model of a five-bar linkage robot with flexible joints and derives a state-space model for the robot as well. In Sections 3 and 4, we describe the algorithmic details of the EKF and UKF formulations and implement these filters for the five-bar linkage robot. Section 5 shows the simulation results and discusses their significance. And Section 6 gives the concluding remarks.

2. Dynamic Modeling

Referring to Figure 1, we define 𝑞₁=[𝑞_𝑙1,…,𝑞_𝑙𝑛]^𝑇 as the vector of link angles and 𝑞₂=[(1/𝑟_𝑚1)𝑞_𝑚1,…,(1/𝑟_𝑚𝑛)𝑞_𝑚𝑛]^𝑇 as the vector of motor shaft angles (reflected to the link side of the gears) for the 𝑛-link flexible-joint manipulator. The dynamic model can be derived using the Euler-Lagrange equations [7, 12]𝐷𝑞₁𝑞₁̈+𝐽𝐶𝑞₂̈𝑞₁,𝑞₁̇𝑞₁̇+𝑔𝑞₁+𝐾𝑞₁−𝑞₂=0,+𝐵𝑞₂̇+𝐾𝑞₂−𝑞₁=𝑢,(2.1) where 𝐷(𝑞₁) is the inertia matrix (symmetric and positive definite), 𝐶(𝑞₁,𝑞₁̇) is the vector representing the damping, Coriolis and centrifugal torques, 𝑔(𝑞₁) is the vector of torques due to gravity, 𝐾=diag[𝑘₁,…,𝑘_𝑛] is the joint stiffness coefficients modeling the joints elasticity, 𝐽=diag[𝐽₁,…,𝐽_𝑛] and 𝐵=diag[𝐵₁,…,𝐵_𝑛] are diagonal matrices representing rotor inertia and rotor damping, respectively, and 𝑢∈𝑅^𝑛 is the vector of input torques applied to the rotor [7, 12].

Figure 1 

Model of a single-link, flexible-joint manipulator.

Figure 2, shows the five-bar linkage manipulator with flexible joints built in robotics research lab in our department. Also, Figure 3 depicts the 5-bar linkage manipulator schematic where the links form a parallelogram [13]. It is clear from this figure that even though there are four links being moved, there are in fact only two degrees of freedom, identified as 𝑞_𝑙1 and 𝑞_𝑙2 [12].

Figure 2 

Five-bar linkage manipulator.

Figure 3 

Planar schematic of 5-bar linkage robot.

We adopt a similar approach, that is, successive differentiation of the link position with respect to time, introduced by [7] to derive a state-space model for the 5-bar linkage robot. So the state vector is defined as 𝑥⃗=𝑞₁𝑞₂𝑞₁̇𝑞₂̇𝑞₁̈𝑞₁⃛^𝑇,(2.2) where 𝑞₁=𝑞_𝑙1𝑞_𝑙2^𝑇,𝑞₂=1𝑟_𝑚1𝑞_𝑚11𝑟_𝑚2𝑞_𝑚2^𝑇.(2.3) Then following the discussion in [12], we set 𝑚₃𝑙₂𝑙_𝑐3=𝑚₄𝑙₁𝑙_𝑐4,(2.4) which subsequently makes 𝑑₁₂ and 𝑑₂₁ be zero, that is, the inertia matrix becomes diagonal and constant. Therefore using (2.1)–(2.4), the rather complex-looking manipulator in Figure 2 can be expressed by the following decoupled set of equations:𝑥𝑥̇̇₁₂=𝑥₅,𝑥̇₃=𝑥₆,𝑥̇₄=𝑥₇,𝑥̇₅=𝑥₈,=1𝑑₁₁−𝑔cos𝑥₁𝑚₁𝑥̇₆𝑙_𝑐1+𝑚₃𝑙_𝑐3+𝑚₄𝑙₁−𝑘₁𝑥₁−𝑥₃,=1𝑑₂₂−𝑔cos𝑥₂𝑚₂𝑥̇₇𝑙_𝑐2−𝑚₄𝑙_𝑐4+𝑚₃𝑙₂−𝑘₁𝑥₂−𝑥₄,=1𝐽₁𝑢₁𝑥̇₈−𝐵₁𝑥₇−𝑘₁𝑥₃−𝑥₁,=1𝐽₂𝑢₂𝑥̇₉−𝐵₂𝑥₈−𝑘₂𝑥₄−𝑥₂,=1𝑑₁₁𝑔𝑥₅𝑥̇₁₀sin𝑥₁𝑚₁𝑙_𝑐1+𝑚₃𝑙_𝑐3+𝑚₄𝑙₁−𝑘₁𝑥₅−𝑥₇,=1𝑑₂₂𝑔𝑥₆𝑥̇₁₁sin𝑥₂𝑚₂𝑙_𝑐2−𝑚₄𝑙_𝑐4+𝑚₃𝑙₂−𝑘₂𝑥₆−𝑥₈,=1𝑑₁₁𝑔𝑥̇₅sin𝑥₁𝑚₁𝑙_𝑐1+𝑚₃𝑙_𝑐3+𝑚₄𝑙₁+𝑔𝑥²⁵cos𝑥₁𝑥̇₁₂×𝑚₁𝑙_𝑐1+𝑚₃𝑙_𝑐3+𝑚₄𝑙₁−𝑘₁𝑥̇₅−𝑥̇₇,=1𝑑₂₂𝑔𝑥̇₆sin𝑥₂𝑚₂𝑙_𝑐2−𝑚₄𝑙_𝑐4+𝑚₃𝑙₂+𝑔𝑥²⁶cos𝑥₂×𝑚₂𝑙_𝑐2−𝑚₄𝑙_𝑐4+𝑚₃𝑙₂−𝑘₂𝑥̇₆−𝑥̇₈.(2.5) This special feature helps to explain the popularity of the parallelogram configuration in industrial robots; since one can adjust the two set of angles (𝑞_𝑙1,𝑞_𝑚1) and (𝑞_𝑙2,𝑞_𝑚2) independently, without worrying about interactions between them.

3. Extended Kalman Filter

The Kalman filter addresses the general problem of trying to estimate the state 𝑥∈𝑅^𝑛 of a discrete-time controlled process that is governed by a linear stochastic difference equation. As an extension to the same idea, the extended Kalman filter (EKF) is used if the dynamic of the system and/or the output dynamic is nonlinear. EKF is based on linearization about the current estimation error mean and covariance [8].

3.1. Definitions

Let us assume that the process has a state vector 𝑥∈𝑅^𝑛 and a control vector 𝑢 and is governed by the nonlinear stochastic difference equation 𝑥_𝑘=𝑓𝑥_𝑘−1,𝑢_𝑘,𝑤_𝑘−1,(3.1) with a measurement 𝑧∈𝑅^𝑚that is 𝑧_𝑘=ℎ𝑥_𝑘,𝑣_𝑘,(3.2) the random variables 𝑤_𝑘 and 𝑣_𝑘represent the process and measurement noise, respectively. They are assumed to be independent of each other, white, and with normal probability distributions with covariance matrices 𝑄 and 𝑅. It can be shown that the time update equations of EKF is 𝑃𝑥̂^−𝑘=𝑓𝑥̂^−𝑘−1,𝑢_𝑘,0_−𝑘,=𝐴_𝑘𝑃_𝑘−1𝐴^𝑇𝑘+𝑊_𝑘𝑄_𝑘−1𝑊^𝑇𝑘,(3.3) where 𝑥̂^−𝑘is the a priori state estimate [8]. These time update equations project the state and covariance estimate (𝑃_𝑘) from the previous time step 𝑘−1to the current time step𝑘. And the measurement update equations of the EKF are 𝐾_𝑘=𝑥̂_𝑘𝑃^−𝑘𝐻^𝑇𝑘𝐻_𝑘𝑃^−𝑘𝐻^𝑇𝑘+𝑉_𝑘𝑅_𝑘𝑉^𝑇𝑘⁻¹,=𝑃𝑥̂^−𝑘+𝐾_𝑘𝑧_𝑘−ℎ𝑥̂^−𝑘,0_𝑘,=𝐼−𝐾_𝑘𝐻_𝑘𝑃^−𝑘,(3.4) where 𝐴,𝑊,𝐻and 𝑉are Jacobian matrices and 𝐾is the correction gain vector. These measurement update equations correct the state and covariance estimate with the measurement 𝑧_𝑘 [8]. The design process of this filter is explained next.

3.2. Implementation

The differential equations are integrated using a fourth-order Runge Kutta method with a step size of 14 msec.

Suppose that the position and velocity of the 5-bar linkage robot are measured as 𝑧_𝑘=⎡⎢⎢⎢⎢⎣𝑞^𝑙1𝑞_𝑙2𝑞̇_𝑙1𝑞̇_𝑙2⎤⎥⎥⎥⎥⎦_𝑘+𝑣_𝑘,(3.5) where 𝑣_𝑘represents the measurement noise.

Due to the recursive nature of the EKF algorithm, the state vector needs to be initialized in startup. The initial position and velocity components are taken as the first measured values. Here, the following initial conditions are selected randomly for the state vector:𝑥_initial=𝜋2𝜋𝜋4𝜋200000000^𝑇.(3.6) We add uncertainty to the initial condition by selecting 𝑃₀=3𝜋180²138𝜋0²128𝜋0²128𝜋0²11111111^𝑇,(3.7) and the process noise and measurement noise are chosen as𝑄=diag3𝜋180²138𝜋0²128𝜋0²128𝜋0²𝑅=diag5𝜋1010202030304040,180²158𝜋0²44.(3.8)

Thus, we developed all the necessary elements of the EKF. In Section 5 the results of simulating the filter are presented.

4. Unscented Kalman Filter

The basic premise behind the unscented Kalman filter is based on the idea that it is easier to approximate a Gaussian distribution than it is to approximate an arbitrary nonlinear function. Instead of linearizing using Jacobian matrices, the UKF uses a deterministic sampling approach to capture the mean and covariance estimates with a minimal set of sample points, and it has 3rd-order (Taylor series expansion) accuracy for Gaussian error distribution for any nonlinear system [11], while EKF uses linearizing Jacobian matrix, which is a first-order approximation. The UKF is claimed to have obvious advantages over EKF [10]. A brief overview of the UKF algorithm is presented in the following section.

4.1. Definitions

The unscented transformation (UT) is a method for calculating the statistics of a random variable which undergoes a nonlinear transformation. The 𝐿 dimensional random variable 𝑥 with mean 𝑥and covariance 𝑃_𝑥is approximated by 2𝐿+1 weighted points given by𝜒𝜒₀=𝑥,𝑖=0,_𝑖𝜒=𝑥+√(𝐿+𝜆)𝑃_𝑥_𝑖,𝑖=1,…,𝐿,_𝑖=𝑥−√(𝐿+𝜆)𝑃_𝑥_𝑖−𝐿,𝑖=𝐿+1,…,2𝐿.(4.1) These sigma points are propagated through the nonlinear function 𝑦_𝑖=𝑓𝜒_𝑖,𝑖=0,…,2𝐿,(4.2) from which the mean and covariance of the transformed probability can be approximated,𝑦≈^2𝐿_𝑖=0𝑃_𝑦𝑊^𝑚𝑖𝑦_𝑖,≈^2𝐿_𝑖=0𝑊^𝑐𝑖𝑦_𝑖−𝑦𝑦_𝑖−𝑦^𝑇,(4.3) with weights 𝑊_𝑖 given by𝑊^𝑚0=𝜆𝑊𝐿+𝜆_𝑐0,=𝜆𝐿+𝜆+1−𝛼²𝑊+𝛽_𝑚𝑖,=𝑊^𝑐𝑖=12(𝐿+𝜆),𝑖=1,…,2𝐿,(4.4) where 𝜆=𝛼²(𝐿+𝜅)−𝐿 is a scaling parameter. The constant 𝛼 determines the spread of the sigma points around 𝑥 and is usually set to small positive values less than one (typically in the range 0.001 to 1) whereas 𝜅 is the secondary scaling parameter usually set to zero or 3−𝐿, and the constant 𝛽 is used to incorporate prior knowledge of the distribution of 𝑥 (for Gaussian distributions, 𝛽=2 is optimal). The scale parameters may be tuned to match the specific problem; some guidelines to choose them are provided in [10].

The unscented Kalman filter (UKF) can be implemented using UT by expanding the state space to include the noise component:𝑥^𝑎𝑘=[𝑥^𝑇𝑘,𝑤^𝑇𝑘,𝑣^𝑇𝑘]^𝑇. The UKF algorithm can be summarized as follows:(1)initialization: 𝑃𝑥_𝑎0̂^𝑎0=𝑥̂^𝑇000^𝑇,=⎡⎢⎢⎣𝑃⁰00𝑄00𝑅00⎤⎥⎥⎦.(4.5)(2)iteration for each time step 𝑘(∈1,…,∞),(a)calculate sigma-points: 𝜒^𝑎𝑘−1=𝑥̂^𝑎𝑘−1𝑥̂^𝑎𝑘−1+𝛾√𝑃^𝑎𝑘−1𝑥̂^𝑎𝑘−1−𝛾√𝑃^𝑎𝑘−1,(4.6)(b)time update: 𝜒𝑥^{𝑥𝑘|𝑘−1}̂_−𝑘=𝑓𝜒^𝑥𝑘−1,𝜒^𝑤𝑘−1,𝑢_𝑘−1,=^2𝐿_𝑖=0𝑃_−𝑘𝑊^𝑚𝑖𝜒^{𝑥𝑖,𝑘∣𝑘−1},=^2𝐿_𝑖=0𝒵𝑊^𝑐𝑖𝜒^{𝑥𝑖,𝑘∣𝑘−1}_{𝑘∣𝑘−1}−𝑥̂^−𝑘𝜒^{𝑥𝑖,𝑘∣𝑘−1}−𝑥̂^−𝑘^𝑇,𝑧_−𝑘̂=ℎ𝜒^{𝑥𝑘∣𝑘−1},𝜒^𝑣𝑘−1,=^2𝐿_𝑖=0𝑊^𝑚𝑖𝒵_{𝑖,𝑘∣𝑘−1};(4.7)(c)measurement update: 𝑃𝑧_𝑘̂𝑧_𝑘̂=^2𝐿_𝑖=0𝑃𝑥̂_𝑘𝑊^𝑐𝑖𝒵_{𝑖,𝑘∣𝑘−1}−𝑧^−𝑘̂𝒵_{𝑖,𝑘∣𝑘−1}−𝑧^−𝑘̂^𝑇,𝑧_𝑘̂=^2𝐿_𝑖=0𝑊^𝑐𝑖𝐾𝜒^{𝑥𝑖,𝑘∣𝑘−1}_𝑘−𝑥̂^−𝑘𝒵^{𝑖,𝑘∣𝑘−1}−𝑧^−𝑘̂^𝑇,=𝑃𝑥̂_𝑘𝑧_𝑘̂𝑃⁻¹𝑧_𝑘̂𝑧_𝑘̂𝑥̂_𝑘,𝑃_𝑘=𝑥̂^−𝑘+𝐾_𝑘𝑧_𝑘−𝑧^−𝑘̂,=𝑃^−𝑘−𝐾_𝑘𝑃𝑧_𝑘̂𝑧_𝑘̂𝐾^𝑇𝑘,(4.8) where 𝜒^𝑎=(𝜒^𝑥)^𝑇(𝜒^𝑤)^𝑇(𝜒^𝑣)^𝑇^𝑇,𝛾=√𝐿+𝜆.(4.9)

4.2. Implementation

The differential equations are integrated using a fourth-order Runge Kutta method with a step size of 14 msec. We initialize the filter in the same way as the EKF, using the same values for the state vector and covariance matrices. We also need to set the tuning parameters𝛼, 𝛽 and 𝜅. The optimum values for coefficients 𝛼 and 𝛽are chosen as 0.001 and 2, respectively. And 𝜅 is set to zero. These optimum values are chosen such that they provide the best estimates for all experiments [11].

5. Simulation Results

This section presents simulation results by Matlab. The simulation data and nominal values of the five-bar linkage parameters are selected as shown in Tables 1 and 2. Also, the simulation step time is chosen 14 msec. To evaluate the performances of the proposed EKF and UKF for the five-bar linkage robot, we plotted the estimated states from the available measurements in Figures 4, 5, 6, and 7. They belong to the the first joint. The results for the second joint are similar.

Table 1 

5-Bar linkage manipulator data.

Table 2 

Simulation data for 5-bar linkage.

Figure 4 

The estimated position 𝑞𝑙1.

Figure 5 

The estimated position 𝑞𝑚1.

Figure 6 

The estimated velocity 𝑞̇𝑙1.

Figure 7 

The estimated velocity 𝑞̇𝑚1.

Moreover, Figures 8, 9, 10, and 11, depict the estimated accelerations and jerks for both of the joints.

Figure 8 

The estimated acceleration 𝑞̈𝑙1 of the first joint.

Figure 9 

The estimated acceleration 𝑞̈𝑙2 of the second joint.

Figure 10 

The estimated jerk 𝑞⃛𝑙1 of the first joint.

Figure 11 

The estimated jerk 𝑞⃛𝑙2 of the second joint.

6. Conclusion

In this paper the extended and unscented Kalman filters are employed for state estimation of a flexible-joint robot. First, the dynamic model of a five-bar manipulator is derived in order to apply the proposed filters. Then, simulation results illustrate that both filters can estimate the link acceleration and provide estimates of the link jerk using the position and velocity measurements. Knowledge of these states is necessary for application of robust outer-loop control theory. In the future we will try to utilize these estimations in robust nonlinear tracking controller design for flexible-joint robots. Furthermore, developing system identification techniques for the five-bar linkage robot would be another challenging task.

Acknowledgment

The authors would like to thanks Ms. Somayyeh Nalan for her assistance.