robot systems to accomplish given tasks in dynam- ic environments. Many vision researchers have been adopting deliberative approaches in order to describe.
IEEE/RSJ/GI International Conference on Intelligent Robots and Systems 1994 (IROS ’94), pp.186–193, M¨ unchen, Sep. 12–16, 1994
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Versatile Visual Servoing without Knowledge of True Jacobian Koh HOSODA and Minoru ASADA Dept. of Mechanical Engineering for Computer-Controlled Machinery, Faculty of Engineering, Osaka University, Suita, Osaka 565, Japan Abstract In this paper, we propose a versatile visual servoing control scheme with a Jacobian matrix estimator. The Jacobian matrix estimator does not need a priori knowledge of the kinematic structure and parameters of the robot system, such as camera and link parameters. The proposed visual servoing control scheme ensures the convergence of the image-features to desired trajectories, by using the estimated Jacobian matrix, which is proved by the Lyapunov stability theory. To show the effectiveness of the proposed scheme, simulation and experimental results are presented.
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control scheme with a Jacobian matrix estimator. It has the following features: 1. The Jacobian matrix estimator does not need any a priori knowledge of the kinematic structure or the system parameters, as far as all the variables of the system is measurable. So we can eliminate the exhausting calibration process. 2. There are no restrictions: number of cameras, camera in hand or camera and hand, SISO or MIMO. The proposed method is applicable to all cases.
Introduction
The use of visual information is indispensable for robot systems to accomplish given tasks in dynamic environments. Many vision researchers have been adopting deliberative approaches in order to describe 3-D scene structure, which are very time consuming, and therefore utilizing these methods to real robot applications seems hard. Recently, there have been many studies on visual servoing control schemes utilizing the visual sensors to increase the capability of the robot systems in the dynamic environments [1]. Here the robot system consists of the control scheme, kinematic structure and system parameters such as link and camera parameters. In these days, many studies focus on control schemes which make features on the image plane converge to the desired values [2–9]. Most of the previous works on visual servoing assume that the system structure and the system parameters are known, or the parameters can be identified in an off-line process (for example,[10]). A control scheme with off-line parameter identification is not robust for disturbance, change of parameters, and unknown environments. To overcome such defects, some on-line parameter identification schemes are proposed [6–9]. Weiss et al. [6] assumed that the system can be modeled by the linear and single input/ single output (SISO) equations, and applied independent MRAC(Model Reference Adaptive Control) controllers. Feddema et al. [7] and Papanikolopoulos et al. [8] modeled the system making use of ARMAX (autoregressive with external inputs) model and estimated the coefficients of the model. Papanikolopoulos et al. [9] estimated the depth related parameters. In these approaches, the structure of the system has to be known. These methods are only applicable to a restricted case, for example, a single camera case. In this paper, we propose a versatile visual servoing
3. The aim of the Jacobian matrix estimator is not to estimate the true parameters, but to ensure asymptotical convergence of the image-features to the desired values under the proposed control scheme. Therefore the estimated parameters do not necessarily converge to the true values. Because of these features, the control scheme does not need to care the complexity of the system structure, the number of cameras, nor the situations of cameras (whether in-hand or fixed to the world coordinate frame). In this sense, we call the proposed scheme a versatile control scheme. This paper is organized as follows. First, from an equation that the estimated Jacobian matrix should hold, a Jacobian matrix estimator is derived in continuous-time domain and then, modified for discrete-time domain. Another estimator is proposed based on the extended least squares algorithm, and is compared with the modified discrete-time estimator. These two kinds of estimator are unified. Then a versatile visual servoing control scheme is proposed including the feedforward term, using the estimated Jacobian matrix. The asymptotical convergence of the image-features to the time-variant desired values is proved by the Lyapunov stability theory. Finally simulation and experimental results show the effectiveness of the proposed scheme.
2 2.1
Real-time Jacobian Matrix Estimator Jacobian Matrix
Only one assumption the proposed servoing control scheme needs is that variables which describe the system displacement are all measurable (See figure 1)． Using the information from sensors, we construct a
and then desired trajectory
xd
x i+1
(5) ˙b The derivative J (t) which satisfies eq.(5) cannot be determined uniquely. We choose one of the solutions of this equation:
x i+2
xi camera camera
i
(i+1)
image
camera
¨ b˙ θ˙ = x b θ. ¨ −J J
(i+2)
image
image
b˙ J b˙ J
=
¨ θ˙ T W (t) b θ) (¨ x−J , T θ˙ W (t)θ˙
(k θ˙ k6= 0),
=
O,
(k θ˙ k= 0),
(6)
where W (t) is a full rank weighting matrix. This solution can be obtained by applying a pseudo-inverse matrix of θ˙ to eq.(5). Formally discretizing eq.(5) with sampling rate ∆t, we get
n
2
b (t) − J b (t − ∆t) = J b (t − ∆t)∆θ(t)}∆θ(t)T W (t) , {∆x(t) − J ∆θ(t)T W (t)∆θ(t)
1
(7)
where ∆θ(t) = θ(t) − θ(t − ∆t) and ∆x(t) = x(t) − x(t − ∆t).
2.3 Figure 1: Robot system equipped with visual sensors
On the other hand, when the system moves slowly, we can use the least squares algorithm with exponential data weighting for estimating the Jacobian matrix. From the formulation of the algorithm, we get
Jacobian matrix estimator. Let θ ∈
