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A model-based approach to robot fault diagnosis
Knowledge-Based Systems 18 (2005) 225–233 www.elsevier.com/locate/knosys
A model-based approach to robot fault diagnosis Honghai Liu*, George M. Coghill Department of Computing Science, University of Aberdeen, Aberdeen AB24 3UE, Scotland, UK Received 26 October 2004; accepted 30 October 2004 Available online 4 June 2005
Abstract This paper presents a model-based approach to online robotic fault diagnosis: First Priority Diagnostic Engine (FPDE). The first principle of FPDE is that a robot is assumed to work well as long as its key variables are within an acceptable range. FPDE consists of four modules: the bounds generator, interval filter, component-based fault reasoner (core of FPDE) and fault reaction. The bounds generator calculates bounds of robot parameters based on interval computation and manufacturing standards. The interval filter provides characteristic values in each predetermined interval to denote corresponding faults. The core of FPDE carries out a two-stage diagnostic process: first it detects whether a robot is faulty by checking the relevant parameters of its end-effector, if a fault is detected it then narrows down the fault at the component level. FPDE can identify single and multiple faults by the introduction of characteristic values. Fault reaction provides an interface to invoke emergency operation or tolerant control, even possibly system reconfiguration. The paper ends with a presentation of simulation results and discussion of a case study. q 2005 Elsevier B.V. All rights reserved. Keywords: Fault diagnosis; Model-based reasoning; Robotics
1. Introduction The problem of robotic diagnosis and fault tolerance presents a considerable challenge to both the artificial intelligence and robotics research communities [1,2]. Many contributions have been made to this topic in the past two decades [3–5]. For example, Visinsky et al. [6] provided a layered fault tolerance framework, for remote robots, consisting of servo, interface and supervisor layers. The layers form a hierarchy of fault tolerance which provide different levels of detection and tolerance capabilities for structurally diverse robots. Schroder [7] proposed a qualitative approach to fault diagnosis of dynamical systems, mainly process control systems. However, most of current fault diagnosis approaches focus on one of robot fault categories, hardware failure, or faults caused by modelling errors or uncertainty. This paper proposes the First Priority Diagnostic Engine, FPDE, as a means to diagnose robot faults from the * Corresponding author. Tel.: C44 1224 274449; fax: C44 1224 273422. E-mail addresses: [email protected] (H. Liu), [email protected]. ac.uk (G.M. Coghill).
0950-7051/$ - see front matter q 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2004.10.004
viewpoint of composite data streams of robotic key variables. The FPDE approach recategorizes robot faults into sensor faults, robotic behaviour faults and modelling errors. Sensor faults can be diagnosed by low-level sensor diagnosis. A robotic behaviour faults are composed of orientation faults and translational faults. The behaviour fault could be caused by hardware failures, e.g. a gearbox fault, excluding sensory fault, and uncertainty factors. Uncertain motion collision can lead to orientation faults and/or translational faults. The FPDE first diagnoses predetermined key priority variables only (e.g. position of an end-effector); a robot is assumed to have no fault if the key priority variables are within acceptable bounds; otherwise it goes to diagnose its variables at a lower level. An interval filter is introduced to deal with noise from measurements. The FPDE deals with a large degree of noise produced from payload changes, by simply ignoring false faults in the time interval of gripper action. Once a fault is detected, characteristic values automatically isolate the fault position. The characteristic mapping presents the relation between the inputs and outputs of a physical system using characteristic values, which are quantities extracted from the quantitative intervals of a domain in order to describe the corresponding qualitative information, for diagnostic purposes. Hence, the FPDE can be applied to general
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Notation q^i , q_^i ith estimated orientation angle and its velocity. Qd, Q_ d desired joint trajectory and its velocity of a robot. Qb, Q_ b bounds of a joint trajectory and its velocity. Qoffset joint repeatability. Qs, Q_ s joint trajectory from sensors and its velocity. QG upper and lower trajectory bounds based on a kinematic model. G Q~ upper and lower trajectory bounds based on a dynamic model. Q^ b , Q^_ b estimated bounds of a joint trajectory and its velocity Q^ s , Q^_ s estimated joint trajectory and its joint speed Q^ i , Q^_ i ith estimated joint trajectory and velocity, respectively
dynamical systems independent by their control systems design. This approach can diagnose both sensor-based parameters and non-sensor-based. In Section 2, we present the problem formulation for robot fault diagnosis and a robotic diagnosis system. In Section 3, we describe the FPDE approach, and in Section 4, we give a case study based on the simplified robot arm of the Beagle 2 Lander, finally conclusions are presented.
t CIi CVj CIiv CVjv Ji l^i ^ Y^ X, ^_ Y^_ X,
computed torque of a robot time instant of a characteristic value of a position interval in the i-coordinate characteristic value of a position interval in the j-coordinate time instant of a characteristic value of a velocity interval in the i-coordinate characteristic value of a velocity interval in the j coordinate joint fault for the i coordinate estimated length of the ith link. estimated position of an end-effector in x-, y-coordinates estimated speed of an end-effector in x-, y-coordinates
cQDti ;
C ðQDti 2½qK Dti ; qDti Þ
† For non-sensor based parameters (e.g. YDti ), to check C whether the following formula is true provided qK Dti , qDti K C and yDti , yDti C cQDti 2½qK Dti ; qDti ;
d fm ðQ1Dti ; .; Qm Dti Þ
C 2½yK Dti ; yDti
2. Problem formulation and solution This section introduces an interval-computation based description for robotic faults, and proposes a model-based reasoning solution to the problem. 2.1. Robotic fault description Generally speaking, there are two types of parameter in robotic diagnosis, sensor-based parameters and non-sensorbased parameters. The latter usually can be mathematically described by the former. Let us use capital letters to describe variables that are not necessarily uniquely determined by our knowledge, i.e. that can take values from an interval. In these terms, if a measurement leads us to a conclusion that C the value of this variable belongs to an interval ½qK Dti ; qDti , K then we can express this knowledge as: ðqDti % QDti % qC Dti Þ. This means there is no fault for variable QDti , otherwise a fault is detected. In these terms, the basic problem of fault diagnosis for a n-link robot can be reformulated into the following two steps: † For sensor based parameters (e.g. QDti ), to check whether the following formula is true provided QDti :
(1)
(2)
where iZ1,2,.,n stand for n sensor-based parameters, e.g. orientation angles from robotic sensors, jZ1,2,.,m stand for m non-sensor-based parameters, e.g. position of an end-effector. Actually by checking the validity of these formulas, we can check whether both sensor based and non-sensor based parameters are within their bounds. If all parameters meet the above bounds requirement, it means that the n-link robot works in an acceptable manner, otherwise a fault has been detected, based on which further diagnosis should be carried out. 2.2. Robot diagnosis system A robot diagnosis system, depicted schematically in Fig. 1, is proposed as a solution to the problem of robotic faults. The structure consists of three parts: the robotic system, motion planner and robotic fault reasoning: the FPDE. The robotic system could be a kinematics-based or dynamics-based system, which usually includes physical robots, sensors and motion control unit. The motion planner is used to plan robot motion to meet the task requirements. It generates the desired data (e.g. Qd, Q_ d and t) for both the motion control unit and fault diagnosis. The motion planner is usually packaged with the robot by the robot provider or
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Motion Planner (Inverse Kinematics)
227
cm
τ . .. Θd Θd Θd
. .. Θd Θd Θd
Motion Control
Bounds Generators
cr
. Θb Θb
Interval Filter . Θb Θb
Interval Filter
. ΘsΘs
Robot
Sensors
Robotic System
. ΘsΘs
Faults Reaction
Component-Based Robotic Reasoning
Faults Report
Fig. 1. The FPDE robotic fault reasoning.
developed by the robotic application developer. It is the main reason why we consider the diagnostic program as a package separated from the motion planner. The FPDE basically generates the desired bounds (e.g. Qb, Q_ b ) of robot parameters, and compares them with sensor data (e.g. Qs, Q_ s ) based on both robotic kinematics or dynamics and interval computation [8]. The outputs of the FPDE are the fault report and fault reaction. The former lists faults and possible failures caused by the fault, the latter performs robotic emergency service or even reconfiguration of the robotic system.
3. Robotic fault reasoning Generally speaking, robotic faults include hardware failures, modelling errors and uncertainty issues. The sensor failures could occur in to optical encoders, tachometers or relevant electrical circuits. The modelling errors are caused mainly by choosing an unsuitable model, for example, a kinematic model being used for robot in a high speed situation. The uncertainty factors usually are from external collisions. The proposed FPDE approach recategorized robotic fault into sensor faults, robotic behaviour faults and modelling errors. A robotic behaviour fault is composed of orientation faults and translational faults. The behaviour fault could be caused by hardware failures, e.g. gearbox faults (excluding sensor fault) and uncertainty factors. Uncertain motion collision can lead to orientation faults and/or translational faults. The operational relation of the FPDE in robotic fault diagnosis is illustrated in Fig. 2. The sensor detection, running first in a diagnostic cycle, is necessary to ensure that the FPDE has a correct scenario. The FPDE approach to robotic fault diagnosis is carried out at the component level of a robot. A robotic component
herein denotes a robotic arm segment (i.e. a link) or a robotic joint (i.e. a motor). The first principle for the FPDE is that a robot operates in acceptable condition if and only if the parameters of its end-effector meet the requirement set by its motion planner. To deal with this, the FPDE is divided into four sections: the bounds generator, interval filter, component-based robotic reasoning (core of the FPDE) and fault remediator. 3.1. Robotic bounds generation Different models affect the fault diagnosis of a robot to some extent [6]. The bounds generator in Fig. 1 is proposed to generate corresponding bounds of the robot parameters depending on different models. In accordance with the diagnostic model of dynamic systems [9], it generates bounds for the end-effector of a robot in the fault detection stage and for each link and joint in the fault isolation stage. In the case of kinematic models [10], the bounds of the C robotic joints, ½qK i ; qi , are easily calculated due to the fact that 90% of their bounds can be determined by the mechanical characteristics of motors [11], Qf, and the manufactured accuracy, Qm. The manufacture accuracy is required to be not more than 5%. Hence, the interval of Q can be set as follows C K C Q Z MaxððqK f % Qf % qf Þ; ðqm % Qm % qm ÞÞ
(3)
where Qf Z ð10=9ÞQoffset , Qm Z 5%Q^ s . The Q is the vector
Sensor Detection
First Priority Diagnostic Engine
Fault Reaction
Fig. 2. Operational relation in robotic diagnostic system.
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consisting of maximum value of Qf, and Qm in each time interval. The Qoffset is robotic repeatability required in accordance with the robot datasheet. The bounds of the robotic parameters of an end-effector (e.g. [yK, yC]) are usually obtained by empirical approaches or experimental methods in practice. 3.2. The interval filter Many theoretical and practical contributions to filter development have been made in both the control and AI communities such as, Kalman filters, extended Kalman filters [12,13], and non-Kalman filters [14,15]. In general, a filter is a device or an algorithm that removes something from whatever passes through it. Filters and filter-related methods extract similarity or characteristics of data from channels to meet system requirement [16,17]. The interval filter is designed to detect faults and extract characteristic values of robot parameters in each predetermined time interval, s. Probability theory is used to detect faults from data with noise. It is a practical problem for diagnosis task to deal with measurements with, sometimes, large amounts of noise. The selection of characteristic values is application dependent. The characteristic values are extracted from time intervals of robotic motion to describe their corresponding fault information with high probability. That is, the fault of a dynamic system is qualitatively described by discrete quantitative values in terms of predefined intervals. A probability function in the time interval j is introduced in the following Ps iZ1 ui P Prðpj Þ Z s K siZ1 ui where ( ui Z 1;
n X iZ1
^ fcp ðQ^ 1 ðtÞo/o Q^ n ðtÞÞ/ XðiÞ (5) ^ fcp ðQ^ 1 ðtÞo/o Q^ n ðtÞÞ/ YðiÞ ^ ^ The qualitative state of XðiÞ and YðiÞ depends on the output of the characteristic functions by checking whether Eq. (4) is true. Similarly, the above can apply to velocity of the end-effector as given as follows ^_ Z K XðtÞ
n X
l^i Q^_ i sin Q^ i ðtÞ
iZ1
^_ Z YðtÞ
n X
(6) l^i Q_^ i cos Q^ i ðtÞ
iZ1
where X_^ 2ðx_K; x_CÞ and Y_^ 2ðy_K; y_CÞ ^_ fcv ðQ^_ 1 ðtÞo/o Q^_ n ðtÞÞ/ XðiÞ (7) ^_ fcv ðQ^_ 1 ðtÞo/o Q^_ n ðtÞÞ/ YðiÞ 3.3. Component-based robotic reasoning
ui 2ððeK% E% eCÞo ðeK% 0o 0% eCÞÞ
ui Z 0; otherwise P The siZ1 ui is the number of sampling data within their bounds. Fault detection is based on the fact whether the ratio is less than a constant threshold l. The threshold for different sensors can be set based on empirical approaches and sensor datasheet. Fault detection is based on the fact whether the ratio is less than a constant threshold l. The threshold for different sensors can be set based on empirical approaches and sensor data. The difference, E, between data from the bounds generator and from robot sensors is defined as eKZQCKQs and eCZQsKQK, for a kinematic model; C K eKZ Q~ K Qs and eCZ Qs K Q~ , for a dynamic model. For example, the position of an end-effector at a time instant can be described by the interval filter as given by the following equation ^ Z XðtÞ
where X^ 2½xK; xC and Y^ 2½yK; yC, Eqs. (4) and (5) are applied; and the X^ and Y^ can be estimated by their characteristic values in the time interval. The characteristic function designed to generate characteristic values is a key problem. In this paper, the characteristic function is defined by the following fact: characteristic values are those whose distances are maxima from corresponding bounds. Therefore, qualitative information in predetermined time interval is denoted by their corresponding characteristic values. The qualitative parameter of an end-effector in the ith time interval can be derived as follows
l^i cos Q^ i ðtÞ
^ Z YðtÞ
n X iZ1
l^i sin Q^ i ðtÞ
(4)
The first principle of component-based robotic reasoning, the core of FPDE, is that a robot is not faulty as long as its first-priority parameter is within the acceptable bounds. For a robot, the first-priority parameters are those of its endeffector. The FPDE can narrow down a fault at robotic component level (e.g. links or motors). The algorithm can detect both single faults and multiple faults in a robotic system. The algorithm FPDE is described in the following: FPDE Algorithm (1) Install Td, T_ d as input, lZlvZlpZ1. (2) a PID-based motion planner generates desired joint information, Qd, Q_ d . (3) Bound generators. (a) IF(Kinematic Model), set kZ5% C (i) Desired bounds of the end-effector, ½TK d ; Td , K C ½T_ d ; T_ d TG d Z Td ð1GkÞ;
G T_ d Z T_ d ð1GkÞ
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C _K _C (ii) Desired joint bounds, ½QK d ; Qd , ½Qd ; Qd
QG d Z MaxðQd ð1GkÞo Qoffset Þ; G Q_ d Z Q_ d ð1GkÞ:
(b) IF(Dynamic Model) G ~_ ~G T_G (i) TG d Z T dG d Z T dG; G G (ii) Qd Z Q~ d ; Q_ d Z Q~_ d (4) Interval filter for sensory data. (a) Characteristic values for the position of the end-effector, TcpZfcp(t, l, Ts), where Ts from equations (4) and (5). (b) Characteristic values for the speed of the end-effector, T_ cv Z fcv ðt; l; T_ s Þ, where T_ s is from Eqs. (4) and (7). (5) Component-based fault reasoning. (a) Reasoning about orientation faults. (i) QuZ Qs K QC QdZ Qs K QK d: d; (ii) IF (QuR0 or Qd%0) ^ and Maxðlp ; QdÞ. ^ THEN Calculate Maxðlp ; QuÞ ^ ^ (iii) Generate orientation faults, OFp ðtp ; lp ; Qu ; Qd Þ C K _ _ (iv) QuZ Q_ s K Q_ d ; QdZ Q_ s K Q_ d : _ _ (v) IF (QuR 0 or Qd% 0) ^_ and Maxðl; QdÞ. ^_ THEN Calculate Maxðlv ; QuÞ ^ ^_ _ (vi) Generate orientation faults, OFv ðtv ; lv ; Qu; QdÞ (b) Reasoning about translational faults. (i) ld ðiÞGZ ld ðiÞð1GkÞ (ii) For iZ1 To Length(Qd) For jZ1 To length(Qs) IF (j Not Equal To i) Then ls(j)Zld(j) l^s ðiÞ can be achieved by Eqs. (4), (5) and (7). IF ðl^s ðiÞ 2½ld ðiÞK; ld ðiÞCÞ THEN No transitional faults ELSE set l^u , l^d . (iii) Generate transitional faults, TFðtl ; l; l^u ; l^d Þ (c) Modelling faults. (6) Return NO FAULTS. The FPDE incorporates functions from the bounds generator, interval filter and component-based reasoning. The bounds generator generates bounds for the Cartesian trajectory, joint trajectory and the other relevant parameters. The bounds generator is based on industrial experience to ensure that relevant accuracy is not more than 5%, and it is within its physical requirement such as repeatability of motors. The interval filter generates characteristic values to compare with sensor data in a predetermined time interval. The characteristic function is designed to meet the corresponding application requirement. The advantage of
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the interval filter is that it converts quantitative information into qualitative information based on numerical data. It provides a solution to clear the ambiguity of conversion between quantitative and qualitative information. That is, the mapping between quantitative and qualitative information can be set by some certain characteristic numerical values. It should be noted that the characteristic function used to generate characteristic values has to be carefully designed in order to meet certain requirements of its corresponding qualitative meaning. Compared with the other robot and mechanism diagnostic approaches (e.g. [6]), the FPDE recategorizes fault into behaviour faults and modelling faults based on sensor faults detection. The behaviour fault is composed of orientation fault and translational fault. The former usually is caused by physical failures of motors and links and uncertainty collision, or link overflection; the latter is mainly caused by link overflection. The orientation fault is detected based on upper bound deviations and lower bound deviations as described in the following: ^ or Maxðlp ; QdÞ; ^ IfðQuR 0 or Qd% 0Þ Then Maxðlp ; QuÞ ^_ or Maxðl ; QdÞ: ^_ _ _ IfðQuR 0 or Qd% 0ÞThen Maxðlv ; QuÞ v Results from the mapping of characteristic values provide sufficient information to detect faults and isolate their location. For example, the function for the orientation ^_ QdÞ, ^_ fault, OFv ðtv ; l; Qu; gives the estimated upper bound deviation, estimated lower bound deviation and the corresponding time instant. Faults can be isolated in their corresponding time intervals, multiple faults can be identified by the characteristic values. The introduction of characteristic values not only helps to clear the ambiguity of conversion between quantitative and qualitative information, but also to detect multiple faults in a time interval. An orientation fault can be detected and isolated with the aid of robotic sensors in the FPDE. Transitional fault can be intelligently diagnosed even without information from sensors. As we know there are usually no sensors for industrial robot links. The FPDE assumes that sensor data from motors is believable, and all link parameters except the test link, are replaced by the desired data. The faulty situation of the test link can be checked to see whether it meets the link bounds in Eq. (5) and ðeK% E% eCÞo ðeK% 0o 0% eCÞ, where eKZlCKls and eCZlsKlK. Modelling errors are unavoidable but can be controlled in the design stage of a control system. Table 1 Joint repeatability parameters Joint
Joint 1
Joint 2
Joint 3
Offset
1.10422deg
0.637987deg
K0.756872deg
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4.1. Simulation results
Fig. 3. The robot arm of Beagle 2 Lander.
4. Case study
trajectory speed of the end effector (meter)
trajectory position of the end effector (meter)
A case study of the simplified robot arm of Beagle 2 is addressed for fault diagnosis purpose in this section (Fig. 3). Beagle 2’s robotic arm in the Mars robot mission was originally devised as a means at removing specific scientific instruments and tools from the Lander and deploying them in positions where they can study or obtain samples of the rocks and soil. The arm supports panoramic cameras and deploys crawling mole to gather subsurface soil samples, returning these to the on-board analytic laboratory. The position and speed of the end-effector is determined by key variables. Repeatability (or joint offset) is taken from the datasheet of Beagle 2 [18] (Table 1). 3
Robot diagnosis, generally speaking, includes fault detection, fault isolation and fault identification [9]. The FPDE applies to first three links of the robot arm with 20time interval. Simulation results are shown in Figs. 4–6, respectively: solid-line trajectories describe the test position and speed; dashed-line trajectories describe their corresponding upper bounds and lower bounds; symbols ‘*’ labelled positions describes the characteristic value caused by a discrepancy in the x coordinate, symbols ‘o’ labelled positions describes characteristic value caused by a discrepancy in the y-coordinate. Fault analysis of the endeffector in Cartesian space is presented in Fig. 4, from which it clearly demonstrates a discrepancy between the tested trajectories and their corresponding bounds. The existence of the characteristic value in key priority parameters means the corresponding time interval containing a robotic fault, as highlighted by FPDE. The location of the robot fault is found in the fault isolation stage. In terms of the definition of robot components, the FPDE diagnoses the motors and links of a robot in turn. The fault location is generated by the mapping of the characteristic values of the end-effector. Fig. 5 presents the characteristic values in the joint trajectories corresponding to the trajectories of the endeffector. Fig. 6 presents the characteristic values in the joint speed corresponding to the speed of the end-effector. The figures illustrate faulty time instants with symbols,
tested speed speed bounds Characteristic values of speed X Characteristic values of speed Y
2 1 0 –1 –2 –3
0
0.5
1
1.5
2
2.5 3 time (second)
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5 3 time (second)
3.5
4
4.5
5
4 2 0 –2 –4
Fig. 4. Fault analysis of trajectories of the end-effector.
1st joint trajectory (rad)
10
2nd joint trajectory (rad)
1.5
3rd joint trajectory (rad)
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3
231
Tested trajectory Trajectory bounds Characteristic values of X Characteristic values of Y
5 0 5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5 3 time (second)
3.5
4
4.5
5
1 0.5 0
2 1 0
Fig. 5. Fault analysis of the joint trajectories.
3rd joint speed (rad/s)
2nd joint speed (rad/s)
1st joint speed (rad/s)
respectively. For example, the characteristic values of the first joint in Fig. 5 allows FPDE to deduce that the fault in time interval 12 are caused by the first joint, this fault leads to faulty motion of the end-effector in both x- and y-coordinates. The fault of the third joint in time interval
17 leads to the faulty motion of the end-effector in the y-coordinate. One of the contributions of the FPDE algorithm is to identify multiple faults in a time interval, for example, the speed fault of the end-effector in first time interval is caused by faults of joints 1 and 2.
6 Tested speed Speed bounds Characteristic values of X Characteristic values of Y
4 2 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
2 1 0 1 2 3 3 2 1 0 1 2
time (second)
Fig. 6. Fault analysis of the joint speeds.
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Table 2 Robotic fault analysis at component level CVx
CIy
CVy
CIxv
CVxv
CIyv
CVyv
Jx
Jy
Jxv
Jyv
fault
2 3 4 5 6 7 8 9 10 11
# # # .577 K.05 K.42 # # # # # K2.22
178 178 # # # # # # # # 2500 2750
.524 .524 # # # # # # # # .0312 K1.53
200 200 252 # # # # # # # 2500 2750
.591 .591 .749 # # # # # # # .032 K1.53
53 53 # # # 1250 1431 # # # # 2750
.135 .135 # # # 2.500 2.442 # # # # K1.53
# # # J2 # # # # # # # J1
# # # # # # # # # # # J1
J2 J2 J2 # # # # # # # # #
J1 J2 # # # # # # # # # #
12
2752
K2.22
2752
K1.53
2752
K1.53
2752
K1.53
J1
J1
#
#
13 14 15 16 17 18 19 20
# # # 4000 4201 # 4750 5000
# # # K.09 .365 # 1.125 .984
# # # # # # # 5000
# # # # # # # K.19
# # # # # # 4737 5000
# # # # # # K.81 K.19
# # 3750 3930 4250 4299 4750 5000
# # K2.50 K2.44 K2.04 K1.95 .773 .193
# # # # # # # J3
# # # # # # # J3
# # # # # # J1 J1
# # # # J3 J3 J1 #
J1, J2 My J2, My J2 Mx Mx, Myv Myv # # # My Mxv J1, Mxv, Myv J1, Mxv, Myv # # Myv Mx, Myv J3, Mx J1, J3 J1, Mx J1, J3, Myv
H. Liu, G.M. Coghill / Knowledge-Based Systems 18 (2005) 225–233
CIx # # # 750 1000 1155 # # # # # 2750
1
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4.2. Fault report A fault report is used to conclude detected fault and to generate preparation for further fault reaction such as tolerant control or even system reconfiguration. The fault report of this case study is given in Table 2. The motion process is divided into 20 time intervals, the symbol ‘#’ means all data in corresponding time interval is not faulty because it remains within its corresponding bounds. The last column ‘fault’ provides the reasons for the faults in the corresponding time interval. Where Ji denotes a fault caused by the ith joint, Mj denotes a fault caused by a modelling error in the jth parameter. For an example of the first time interval, there are characteristic values CVy, CVxv, CVyv, which indicates there are faults related to the parameters of the end-effector in the y position coordinate and both x and y velocity coordinates, respectively. Then FPDE checks each component with the provided characteric values. The output of the component level diagnosis is that the characteristic values CVxv, CVyv lead to motor faults, i.e. J1, J2, the characteristic value CVy did not occur any behavior faults but modelling errors, i.e. My.
5. Conclusion The first principle of the FPDE is that a robot is assumed to work well as long as its key variables are within acceptable bounds. Basically, FPDE first calculates characteristic values for key variables of a robot in a predetermined interval time, then maps them into corresponding qualitative states to present their quantitative characteristics. Finally, it compares the characteristic values with their bounds to diagnose faults. The FPDE is composed of four modules: the bounds generator, interval filter, component-based fault reasoning (core of FPDE) and fault reaction. The advantages of the FPDE are the following: It can (1) diagnose both sensor-based parameters and non-sensorbased, (2) detect single faults and multiple faults using characteristic values, (3) provide a sort of measurement of modelling errors to assess the suitability of models. Acknowledgements The financial support from the Engineering and Physical Science Research Council, United Kingdom, under grant
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GR/S10773/01 and GR/S10766/01 is gratefully acknowledged. The authors also would like to thank their partners: Dave Barnes and Andy Shaw from University of Wales, Aberystwyth.
References [1] M.L. Visinsky, J.R. Cavallaro, I.D. Walker, Robotic fault detection and fault tolerance: a survey, Reliability Engineering and System Safety 46 (2) (1994) 139–158. [2] R. Patton, P. Frank, R. Clark, Fault Diagnosis in Dynamic Systems (Theory and Applications), Prentice Hall, Englewood Cliffs, NJ, 1989. [3] J. De Kleer, B.C. Williams, Diagnosing multiple faults, Artificial Intelligence 32 (1987) 100–117. [4] M. Blanke, M. Kinnaert, J. Lunze, M. Staroswiecki, Diagnosis and Fault-Tolerant Control, Springer, Berlin, 2003. [5] U. Heller, P. Struss, gCde: the generalize diagnosis engine, 12th International Workshop on Principles of Diagnosis, 2001. [6] M.L. Visinsky, J.R. Cavallaro, I.D. Walker, A dynamic fault tolerance framework for remote robots, IEEE Transactions on Robotics and Automation 11 (4) (1995) 477–490. [7] J. Schroder, Modelling, State Observation and Diagnosis of Quantised Systems, Springer, Berlin, 2003. [8] B. Bouchon-Meunier, V. Kreinovich, From interval computations to modal mathematics: applications and computational complexity, ACM SIGSAM Bulletin 32 (2) (1998) 7–11. [9] G.M. Coghill, Q. Shen, Towards the specification of models for diagnosis of dynamic systems, Artificial Intelligence in Communications 14 (2) (2001). [10] A. Castellet, F. Thomas, An algorithm for the solution of inverse kinematic problems based on an interval method in: M. Husty, J. Lenarcic (Eds.), Advances in Robot Kinematics, Kluwer Academic Publishers, Dordecht, 1998, pp. 393–403. [11] P.S. Shiakolas, K.L. Conrad, T.C. Yih, On the accuracy, repeatability, and degree of influence of kinematics parameters for industrial robots, International Journal of Modelling and Simulation 22 (3) (2002) 1–10. [12] M.S. Grewal, Kalman Filtering: Theory and Practice, Prentice-Hall, Englewood Cliffs, NJ, 2001. [13] T.W. Parks, C.S. Burrus, Digital Filter Design, Wiley, New York, 1987. [14] P.A. Naylor, O. Tanrkulu, A.G. Constantinides, Subband adaptive filtering for acoustic echo control using allpass polyphase iir filterbanks, IEEE Transactions on Signal and Processing 6 (2) (1998). [15] R. Dearden, D. Clancy, Particle filters for real-time fault detection in planetary rovers, 13th International Workshop on Principles of Diagnosis, 2003. [16] J. Schroder, Modelling, State Observation and Diagnosis of Quantised Systems, Springer, Berlin, 2003. [17] C.S. Perng, H.X. Wang, S.R. Zhang, D.S. Parker, Landmark: a new technique for similarity-based pattern querying in time series databases, Proceedings, ICDE, 2000. [18] D.P. Barnes, E. Taylor, N. Phillips, A. Gossant, G. Paar, Beagle2 simulation, kinematics calibration, and environment dem generation, Proceedings of the Seventh ESA Workshop on Advanced Space Technologies for Robotics and Automation, 2002.
A model-based approach to robot fault diagnosis Honghai Liu*, George M. Coghill Department of Computing Science, University of Aberdeen, Aberdeen AB24 3UE, Scotland, UK Received 26 October 2004; accepted 30 October 2004 Available online 4 June 2005
Abstract This paper presents a model-based approach to online robotic fault diagnosis: First Priority Diagnostic Engine (FPDE). The first principle of FPDE is that a robot is assumed to work well as long as its key variables are within an acceptable range. FPDE consists of four modules: the bounds generator, interval filter, component-based fault reasoner (core of FPDE) and fault reaction. The bounds generator calculates bounds of robot parameters based on interval computation and manufacturing standards. The interval filter provides characteristic values in each predetermined interval to denote corresponding faults. The core of FPDE carries out a two-stage diagnostic process: first it detects whether a robot is faulty by checking the relevant parameters of its end-effector, if a fault is detected it then narrows down the fault at the component level. FPDE can identify single and multiple faults by the introduction of characteristic values. Fault reaction provides an interface to invoke emergency operation or tolerant control, even possibly system reconfiguration. The paper ends with a presentation of simulation results and discussion of a case study. q 2005 Elsevier B.V. All rights reserved. Keywords: Fault diagnosis; Model-based reasoning; Robotics
1. Introduction The problem of robotic diagnosis and fault tolerance presents a considerable challenge to both the artificial intelligence and robotics research communities [1,2]. Many contributions have been made to this topic in the past two decades [3–5]. For example, Visinsky et al. [6] provided a layered fault tolerance framework, for remote robots, consisting of servo, interface and supervisor layers. The layers form a hierarchy of fault tolerance which provide different levels of detection and tolerance capabilities for structurally diverse robots. Schroder [7] proposed a qualitative approach to fault diagnosis of dynamical systems, mainly process control systems. However, most of current fault diagnosis approaches focus on one of robot fault categories, hardware failure, or faults caused by modelling errors or uncertainty. This paper proposes the First Priority Diagnostic Engine, FPDE, as a means to diagnose robot faults from the * Corresponding author. Tel.: C44 1224 274449; fax: C44 1224 273422. E-mail addresses: [email protected] (H. Liu), [email protected]. ac.uk (G.M. Coghill).
0950-7051/$ - see front matter q 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.knosys.2004.10.004
viewpoint of composite data streams of robotic key variables. The FPDE approach recategorizes robot faults into sensor faults, robotic behaviour faults and modelling errors. Sensor faults can be diagnosed by low-level sensor diagnosis. A robotic behaviour faults are composed of orientation faults and translational faults. The behaviour fault could be caused by hardware failures, e.g. a gearbox fault, excluding sensory fault, and uncertainty factors. Uncertain motion collision can lead to orientation faults and/or translational faults. The FPDE first diagnoses predetermined key priority variables only (e.g. position of an end-effector); a robot is assumed to have no fault if the key priority variables are within acceptable bounds; otherwise it goes to diagnose its variables at a lower level. An interval filter is introduced to deal with noise from measurements. The FPDE deals with a large degree of noise produced from payload changes, by simply ignoring false faults in the time interval of gripper action. Once a fault is detected, characteristic values automatically isolate the fault position. The characteristic mapping presents the relation between the inputs and outputs of a physical system using characteristic values, which are quantities extracted from the quantitative intervals of a domain in order to describe the corresponding qualitative information, for diagnostic purposes. Hence, the FPDE can be applied to general
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Notation q^i , q_^i ith estimated orientation angle and its velocity. Qd, Q_ d desired joint trajectory and its velocity of a robot. Qb, Q_ b bounds of a joint trajectory and its velocity. Qoffset joint repeatability. Qs, Q_ s joint trajectory from sensors and its velocity. QG upper and lower trajectory bounds based on a kinematic model. G Q~ upper and lower trajectory bounds based on a dynamic model. Q^ b , Q^_ b estimated bounds of a joint trajectory and its velocity Q^ s , Q^_ s estimated joint trajectory and its joint speed Q^ i , Q^_ i ith estimated joint trajectory and velocity, respectively
dynamical systems independent by their control systems design. This approach can diagnose both sensor-based parameters and non-sensor-based. In Section 2, we present the problem formulation for robot fault diagnosis and a robotic diagnosis system. In Section 3, we describe the FPDE approach, and in Section 4, we give a case study based on the simplified robot arm of the Beagle 2 Lander, finally conclusions are presented.
t CIi CVj CIiv CVjv Ji l^i ^ Y^ X, ^_ Y^_ X,
computed torque of a robot time instant of a characteristic value of a position interval in the i-coordinate characteristic value of a position interval in the j-coordinate time instant of a characteristic value of a velocity interval in the i-coordinate characteristic value of a velocity interval in the j coordinate joint fault for the i coordinate estimated length of the ith link. estimated position of an end-effector in x-, y-coordinates estimated speed of an end-effector in x-, y-coordinates
cQDti ;
C ðQDti 2½qK Dti ; qDti Þ
† For non-sensor based parameters (e.g. YDti ), to check C whether the following formula is true provided qK Dti , qDti K C and yDti , yDti C cQDti 2½qK Dti ; qDti ;
d fm ðQ1Dti ; .; Qm Dti Þ
C 2½yK Dti ; yDti
2. Problem formulation and solution This section introduces an interval-computation based description for robotic faults, and proposes a model-based reasoning solution to the problem. 2.1. Robotic fault description Generally speaking, there are two types of parameter in robotic diagnosis, sensor-based parameters and non-sensorbased parameters. The latter usually can be mathematically described by the former. Let us use capital letters to describe variables that are not necessarily uniquely determined by our knowledge, i.e. that can take values from an interval. In these terms, if a measurement leads us to a conclusion that C the value of this variable belongs to an interval ½qK Dti ; qDti , K then we can express this knowledge as: ðqDti % QDti % qC Dti Þ. This means there is no fault for variable QDti , otherwise a fault is detected. In these terms, the basic problem of fault diagnosis for a n-link robot can be reformulated into the following two steps: † For sensor based parameters (e.g. QDti ), to check whether the following formula is true provided QDti :
(1)
(2)
where iZ1,2,.,n stand for n sensor-based parameters, e.g. orientation angles from robotic sensors, jZ1,2,.,m stand for m non-sensor-based parameters, e.g. position of an end-effector. Actually by checking the validity of these formulas, we can check whether both sensor based and non-sensor based parameters are within their bounds. If all parameters meet the above bounds requirement, it means that the n-link robot works in an acceptable manner, otherwise a fault has been detected, based on which further diagnosis should be carried out. 2.2. Robot diagnosis system A robot diagnosis system, depicted schematically in Fig. 1, is proposed as a solution to the problem of robotic faults. The structure consists of three parts: the robotic system, motion planner and robotic fault reasoning: the FPDE. The robotic system could be a kinematics-based or dynamics-based system, which usually includes physical robots, sensors and motion control unit. The motion planner is used to plan robot motion to meet the task requirements. It generates the desired data (e.g. Qd, Q_ d and t) for both the motion control unit and fault diagnosis. The motion planner is usually packaged with the robot by the robot provider or
H. Liu, G.M. Coghill / Knowledge-Based Systems 18 (2005) 225–233
Motion Planner (Inverse Kinematics)
227
cm
τ . .. Θd Θd Θd
. .. Θd Θd Θd
Motion Control
Bounds Generators
cr
. Θb Θb
Interval Filter . Θb Θb
Interval Filter
. ΘsΘs
Robot
Sensors
Robotic System
. ΘsΘs
Faults Reaction
Component-Based Robotic Reasoning
Faults Report
Fig. 1. The FPDE robotic fault reasoning.
developed by the robotic application developer. It is the main reason why we consider the diagnostic program as a package separated from the motion planner. The FPDE basically generates the desired bounds (e.g. Qb, Q_ b ) of robot parameters, and compares them with sensor data (e.g. Qs, Q_ s ) based on both robotic kinematics or dynamics and interval computation [8]. The outputs of the FPDE are the fault report and fault reaction. The former lists faults and possible failures caused by the fault, the latter performs robotic emergency service or even reconfiguration of the robotic system.
3. Robotic fault reasoning Generally speaking, robotic faults include hardware failures, modelling errors and uncertainty issues. The sensor failures could occur in to optical encoders, tachometers or relevant electrical circuits. The modelling errors are caused mainly by choosing an unsuitable model, for example, a kinematic model being used for robot in a high speed situation. The uncertainty factors usually are from external collisions. The proposed FPDE approach recategorized robotic fault into sensor faults, robotic behaviour faults and modelling errors. A robotic behaviour fault is composed of orientation faults and translational faults. The behaviour fault could be caused by hardware failures, e.g. gearbox faults (excluding sensor fault) and uncertainty factors. Uncertain motion collision can lead to orientation faults and/or translational faults. The operational relation of the FPDE in robotic fault diagnosis is illustrated in Fig. 2. The sensor detection, running first in a diagnostic cycle, is necessary to ensure that the FPDE has a correct scenario. The FPDE approach to robotic fault diagnosis is carried out at the component level of a robot. A robotic component
herein denotes a robotic arm segment (i.e. a link) or a robotic joint (i.e. a motor). The first principle for the FPDE is that a robot operates in acceptable condition if and only if the parameters of its end-effector meet the requirement set by its motion planner. To deal with this, the FPDE is divided into four sections: the bounds generator, interval filter, component-based robotic reasoning (core of the FPDE) and fault remediator. 3.1. Robotic bounds generation Different models affect the fault diagnosis of a robot to some extent [6]. The bounds generator in Fig. 1 is proposed to generate corresponding bounds of the robot parameters depending on different models. In accordance with the diagnostic model of dynamic systems [9], it generates bounds for the end-effector of a robot in the fault detection stage and for each link and joint in the fault isolation stage. In the case of kinematic models [10], the bounds of the C robotic joints, ½qK i ; qi , are easily calculated due to the fact that 90% of their bounds can be determined by the mechanical characteristics of motors [11], Qf, and the manufactured accuracy, Qm. The manufacture accuracy is required to be not more than 5%. Hence, the interval of Q can be set as follows C K C Q Z MaxððqK f % Qf % qf Þ; ðqm % Qm % qm ÞÞ
(3)
where Qf Z ð10=9ÞQoffset , Qm Z 5%Q^ s . The Q is the vector
Sensor Detection
First Priority Diagnostic Engine
Fault Reaction
Fig. 2. Operational relation in robotic diagnostic system.
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consisting of maximum value of Qf, and Qm in each time interval. The Qoffset is robotic repeatability required in accordance with the robot datasheet. The bounds of the robotic parameters of an end-effector (e.g. [yK, yC]) are usually obtained by empirical approaches or experimental methods in practice. 3.2. The interval filter Many theoretical and practical contributions to filter development have been made in both the control and AI communities such as, Kalman filters, extended Kalman filters [12,13], and non-Kalman filters [14,15]. In general, a filter is a device or an algorithm that removes something from whatever passes through it. Filters and filter-related methods extract similarity or characteristics of data from channels to meet system requirement [16,17]. The interval filter is designed to detect faults and extract characteristic values of robot parameters in each predetermined time interval, s. Probability theory is used to detect faults from data with noise. It is a practical problem for diagnosis task to deal with measurements with, sometimes, large amounts of noise. The selection of characteristic values is application dependent. The characteristic values are extracted from time intervals of robotic motion to describe their corresponding fault information with high probability. That is, the fault of a dynamic system is qualitatively described by discrete quantitative values in terms of predefined intervals. A probability function in the time interval j is introduced in the following Ps iZ1 ui P Prðpj Þ Z s K siZ1 ui where ( ui Z 1;
n X iZ1
^ fcp ðQ^ 1 ðtÞo/o Q^ n ðtÞÞ/ XðiÞ (5) ^ fcp ðQ^ 1 ðtÞo/o Q^ n ðtÞÞ/ YðiÞ ^ ^ The qualitative state of XðiÞ and YðiÞ depends on the output of the characteristic functions by checking whether Eq. (4) is true. Similarly, the above can apply to velocity of the end-effector as given as follows ^_ Z K XðtÞ
n X
l^i Q^_ i sin Q^ i ðtÞ
iZ1
^_ Z YðtÞ
n X
(6) l^i Q_^ i cos Q^ i ðtÞ
iZ1
where X_^ 2ðx_K; x_CÞ and Y_^ 2ðy_K; y_CÞ ^_ fcv ðQ^_ 1 ðtÞo/o Q^_ n ðtÞÞ/ XðiÞ (7) ^_ fcv ðQ^_ 1 ðtÞo/o Q^_ n ðtÞÞ/ YðiÞ 3.3. Component-based robotic reasoning
ui 2ððeK% E% eCÞo ðeK% 0o 0% eCÞÞ
ui Z 0; otherwise P The siZ1 ui is the number of sampling data within their bounds. Fault detection is based on the fact whether the ratio is less than a constant threshold l. The threshold for different sensors can be set based on empirical approaches and sensor datasheet. Fault detection is based on the fact whether the ratio is less than a constant threshold l. The threshold for different sensors can be set based on empirical approaches and sensor data. The difference, E, between data from the bounds generator and from robot sensors is defined as eKZQCKQs and eCZQsKQK, for a kinematic model; C K eKZ Q~ K Qs and eCZ Qs K Q~ , for a dynamic model. For example, the position of an end-effector at a time instant can be described by the interval filter as given by the following equation ^ Z XðtÞ
where X^ 2½xK; xC and Y^ 2½yK; yC, Eqs. (4) and (5) are applied; and the X^ and Y^ can be estimated by their characteristic values in the time interval. The characteristic function designed to generate characteristic values is a key problem. In this paper, the characteristic function is defined by the following fact: characteristic values are those whose distances are maxima from corresponding bounds. Therefore, qualitative information in predetermined time interval is denoted by their corresponding characteristic values. The qualitative parameter of an end-effector in the ith time interval can be derived as follows
l^i cos Q^ i ðtÞ
^ Z YðtÞ
n X iZ1
l^i sin Q^ i ðtÞ
(4)
The first principle of component-based robotic reasoning, the core of FPDE, is that a robot is not faulty as long as its first-priority parameter is within the acceptable bounds. For a robot, the first-priority parameters are those of its endeffector. The FPDE can narrow down a fault at robotic component level (e.g. links or motors). The algorithm can detect both single faults and multiple faults in a robotic system. The algorithm FPDE is described in the following: FPDE Algorithm (1) Install Td, T_ d as input, lZlvZlpZ1. (2) a PID-based motion planner generates desired joint information, Qd, Q_ d . (3) Bound generators. (a) IF(Kinematic Model), set kZ5% C (i) Desired bounds of the end-effector, ½TK d ; Td , K C ½T_ d ; T_ d TG d Z Td ð1GkÞ;
G T_ d Z T_ d ð1GkÞ
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C _K _C (ii) Desired joint bounds, ½QK d ; Qd , ½Qd ; Qd
QG d Z MaxðQd ð1GkÞo Qoffset Þ; G Q_ d Z Q_ d ð1GkÞ:
(b) IF(Dynamic Model) G ~_ ~G T_G (i) TG d Z T dG d Z T dG; G G (ii) Qd Z Q~ d ; Q_ d Z Q~_ d (4) Interval filter for sensory data. (a) Characteristic values for the position of the end-effector, TcpZfcp(t, l, Ts), where Ts from equations (4) and (5). (b) Characteristic values for the speed of the end-effector, T_ cv Z fcv ðt; l; T_ s Þ, where T_ s is from Eqs. (4) and (7). (5) Component-based fault reasoning. (a) Reasoning about orientation faults. (i) QuZ Qs K QC QdZ Qs K QK d: d; (ii) IF (QuR0 or Qd%0) ^ and Maxðlp ; QdÞ. ^ THEN Calculate Maxðlp ; QuÞ ^ ^ (iii) Generate orientation faults, OFp ðtp ; lp ; Qu ; Qd Þ C K _ _ (iv) QuZ Q_ s K Q_ d ; QdZ Q_ s K Q_ d : _ _ (v) IF (QuR 0 or Qd% 0) ^_ and Maxðl; QdÞ. ^_ THEN Calculate Maxðlv ; QuÞ ^ ^_ _ (vi) Generate orientation faults, OFv ðtv ; lv ; Qu; QdÞ (b) Reasoning about translational faults. (i) ld ðiÞGZ ld ðiÞð1GkÞ (ii) For iZ1 To Length(Qd) For jZ1 To length(Qs) IF (j Not Equal To i) Then ls(j)Zld(j) l^s ðiÞ can be achieved by Eqs. (4), (5) and (7). IF ðl^s ðiÞ 2½ld ðiÞK; ld ðiÞCÞ THEN No transitional faults ELSE set l^u , l^d . (iii) Generate transitional faults, TFðtl ; l; l^u ; l^d Þ (c) Modelling faults. (6) Return NO FAULTS. The FPDE incorporates functions from the bounds generator, interval filter and component-based reasoning. The bounds generator generates bounds for the Cartesian trajectory, joint trajectory and the other relevant parameters. The bounds generator is based on industrial experience to ensure that relevant accuracy is not more than 5%, and it is within its physical requirement such as repeatability of motors. The interval filter generates characteristic values to compare with sensor data in a predetermined time interval. The characteristic function is designed to meet the corresponding application requirement. The advantage of
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the interval filter is that it converts quantitative information into qualitative information based on numerical data. It provides a solution to clear the ambiguity of conversion between quantitative and qualitative information. That is, the mapping between quantitative and qualitative information can be set by some certain characteristic numerical values. It should be noted that the characteristic function used to generate characteristic values has to be carefully designed in order to meet certain requirements of its corresponding qualitative meaning. Compared with the other robot and mechanism diagnostic approaches (e.g. [6]), the FPDE recategorizes fault into behaviour faults and modelling faults based on sensor faults detection. The behaviour fault is composed of orientation fault and translational fault. The former usually is caused by physical failures of motors and links and uncertainty collision, or link overflection; the latter is mainly caused by link overflection. The orientation fault is detected based on upper bound deviations and lower bound deviations as described in the following: ^ or Maxðlp ; QdÞ; ^ IfðQuR 0 or Qd% 0Þ Then Maxðlp ; QuÞ ^_ or Maxðl ; QdÞ: ^_ _ _ IfðQuR 0 or Qd% 0ÞThen Maxðlv ; QuÞ v Results from the mapping of characteristic values provide sufficient information to detect faults and isolate their location. For example, the function for the orientation ^_ QdÞ, ^_ fault, OFv ðtv ; l; Qu; gives the estimated upper bound deviation, estimated lower bound deviation and the corresponding time instant. Faults can be isolated in their corresponding time intervals, multiple faults can be identified by the characteristic values. The introduction of characteristic values not only helps to clear the ambiguity of conversion between quantitative and qualitative information, but also to detect multiple faults in a time interval. An orientation fault can be detected and isolated with the aid of robotic sensors in the FPDE. Transitional fault can be intelligently diagnosed even without information from sensors. As we know there are usually no sensors for industrial robot links. The FPDE assumes that sensor data from motors is believable, and all link parameters except the test link, are replaced by the desired data. The faulty situation of the test link can be checked to see whether it meets the link bounds in Eq. (5) and ðeK% E% eCÞo ðeK% 0o 0% eCÞ, where eKZlCKls and eCZlsKlK. Modelling errors are unavoidable but can be controlled in the design stage of a control system. Table 1 Joint repeatability parameters Joint
Joint 1
Joint 2
Joint 3
Offset
1.10422deg
0.637987deg
K0.756872deg
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4.1. Simulation results
Fig. 3. The robot arm of Beagle 2 Lander.
4. Case study
trajectory speed of the end effector (meter)
trajectory position of the end effector (meter)
A case study of the simplified robot arm of Beagle 2 is addressed for fault diagnosis purpose in this section (Fig. 3). Beagle 2’s robotic arm in the Mars robot mission was originally devised as a means at removing specific scientific instruments and tools from the Lander and deploying them in positions where they can study or obtain samples of the rocks and soil. The arm supports panoramic cameras and deploys crawling mole to gather subsurface soil samples, returning these to the on-board analytic laboratory. The position and speed of the end-effector is determined by key variables. Repeatability (or joint offset) is taken from the datasheet of Beagle 2 [18] (Table 1). 3
Robot diagnosis, generally speaking, includes fault detection, fault isolation and fault identification [9]. The FPDE applies to first three links of the robot arm with 20time interval. Simulation results are shown in Figs. 4–6, respectively: solid-line trajectories describe the test position and speed; dashed-line trajectories describe their corresponding upper bounds and lower bounds; symbols ‘*’ labelled positions describes the characteristic value caused by a discrepancy in the x coordinate, symbols ‘o’ labelled positions describes characteristic value caused by a discrepancy in the y-coordinate. Fault analysis of the endeffector in Cartesian space is presented in Fig. 4, from which it clearly demonstrates a discrepancy between the tested trajectories and their corresponding bounds. The existence of the characteristic value in key priority parameters means the corresponding time interval containing a robotic fault, as highlighted by FPDE. The location of the robot fault is found in the fault isolation stage. In terms of the definition of robot components, the FPDE diagnoses the motors and links of a robot in turn. The fault location is generated by the mapping of the characteristic values of the end-effector. Fig. 5 presents the characteristic values in the joint trajectories corresponding to the trajectories of the endeffector. Fig. 6 presents the characteristic values in the joint speed corresponding to the speed of the end-effector. The figures illustrate faulty time instants with symbols,
tested speed speed bounds Characteristic values of speed X Characteristic values of speed Y
2 1 0 –1 –2 –3
0
0.5
1
1.5
2
2.5 3 time (second)
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5 3 time (second)
3.5
4
4.5
5
4 2 0 –2 –4
Fig. 4. Fault analysis of trajectories of the end-effector.
1st joint trajectory (rad)
10
2nd joint trajectory (rad)
1.5
3rd joint trajectory (rad)
H. Liu, G.M. Coghill / Knowledge-Based Systems 18 (2005) 225–233
3
231
Tested trajectory Trajectory bounds Characteristic values of X Characteristic values of Y
5 0 5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5 3 time (second)
3.5
4
4.5
5
1 0.5 0
2 1 0
Fig. 5. Fault analysis of the joint trajectories.
3rd joint speed (rad/s)
2nd joint speed (rad/s)
1st joint speed (rad/s)
respectively. For example, the characteristic values of the first joint in Fig. 5 allows FPDE to deduce that the fault in time interval 12 are caused by the first joint, this fault leads to faulty motion of the end-effector in both x- and y-coordinates. The fault of the third joint in time interval
17 leads to the faulty motion of the end-effector in the y-coordinate. One of the contributions of the FPDE algorithm is to identify multiple faults in a time interval, for example, the speed fault of the end-effector in first time interval is caused by faults of joints 1 and 2.
6 Tested speed Speed bounds Characteristic values of X Characteristic values of Y
4 2 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
2 1 0 1 2 3 3 2 1 0 1 2
time (second)
Fig. 6. Fault analysis of the joint speeds.
232
Table 2 Robotic fault analysis at component level CVx
CIy
CVy
CIxv
CVxv
CIyv
CVyv
Jx
Jy
Jxv
Jyv
fault
2 3 4 5 6 7 8 9 10 11
# # # .577 K.05 K.42 # # # # # K2.22
178 178 # # # # # # # # 2500 2750
.524 .524 # # # # # # # # .0312 K1.53
200 200 252 # # # # # # # 2500 2750
.591 .591 .749 # # # # # # # .032 K1.53
53 53 # # # 1250 1431 # # # # 2750
.135 .135 # # # 2.500 2.442 # # # # K1.53
# # # J2 # # # # # # # J1
# # # # # # # # # # # J1
J2 J2 J2 # # # # # # # # #
J1 J2 # # # # # # # # # #
12
2752
K2.22
2752
K1.53
2752
K1.53
2752
K1.53
J1
J1
#
#
13 14 15 16 17 18 19 20
# # # 4000 4201 # 4750 5000
# # # K.09 .365 # 1.125 .984
# # # # # # # 5000
# # # # # # # K.19
# # # # # # 4737 5000
# # # # # # K.81 K.19
# # 3750 3930 4250 4299 4750 5000
# # K2.50 K2.44 K2.04 K1.95 .773 .193
# # # # # # # J3
# # # # # # # J3
# # # # # # J1 J1
# # # # J3 J3 J1 #
J1, J2 My J2, My J2 Mx Mx, Myv Myv # # # My Mxv J1, Mxv, Myv J1, Mxv, Myv # # Myv Mx, Myv J3, Mx J1, J3 J1, Mx J1, J3, Myv
H. Liu, G.M. Coghill / Knowledge-Based Systems 18 (2005) 225–233
CIx # # # 750 1000 1155 # # # # # 2750
1
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4.2. Fault report A fault report is used to conclude detected fault and to generate preparation for further fault reaction such as tolerant control or even system reconfiguration. The fault report of this case study is given in Table 2. The motion process is divided into 20 time intervals, the symbol ‘#’ means all data in corresponding time interval is not faulty because it remains within its corresponding bounds. The last column ‘fault’ provides the reasons for the faults in the corresponding time interval. Where Ji denotes a fault caused by the ith joint, Mj denotes a fault caused by a modelling error in the jth parameter. For an example of the first time interval, there are characteristic values CVy, CVxv, CVyv, which indicates there are faults related to the parameters of the end-effector in the y position coordinate and both x and y velocity coordinates, respectively. Then FPDE checks each component with the provided characteric values. The output of the component level diagnosis is that the characteristic values CVxv, CVyv lead to motor faults, i.e. J1, J2, the characteristic value CVy did not occur any behavior faults but modelling errors, i.e. My.
5. Conclusion The first principle of the FPDE is that a robot is assumed to work well as long as its key variables are within acceptable bounds. Basically, FPDE first calculates characteristic values for key variables of a robot in a predetermined interval time, then maps them into corresponding qualitative states to present their quantitative characteristics. Finally, it compares the characteristic values with their bounds to diagnose faults. The FPDE is composed of four modules: the bounds generator, interval filter, component-based fault reasoning (core of FPDE) and fault reaction. The advantages of the FPDE are the following: It can (1) diagnose both sensor-based parameters and non-sensorbased, (2) detect single faults and multiple faults using characteristic values, (3) provide a sort of measurement of modelling errors to assess the suitability of models. Acknowledgements The financial support from the Engineering and Physical Science Research Council, United Kingdom, under grant
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GR/S10773/01 and GR/S10766/01 is gratefully acknowledged. The authors also would like to thank their partners: Dave Barnes and Andy Shaw from University of Wales, Aberystwyth.
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