Influence of Sensor Quantization on the Control ... - IEEE Xplore
Proceedings of the 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems San Diego, CA, USA, Oct 29 - Nov 2, 2007
TuD2.1
Influence of sensor quantization on the control performance of robotics actuators Renat Iskakov, Alin Albu-Schaeffer, Manfred Schedl, Gerd Hirzinger and Vitaly Lopota* Institute of Robotics and Mechatronics, German Aerospace Center (DLR), Wessling, Germany E-mails: {Renat.Iskakov, Alin.Albu-Schaeffer, Manfred.Schedl, Gerd.Hirzinger}@dlr.de *S.P. Korolev Rocket and Space Corporation - Energia, Korolev, Moscow Region, Russia E-mail: [email protected]
Abstract— In this paper the effect of sensor quantization on the control performance of robotics actuators in the steadystate condition is considered. First, the existence of a limit cycle mode due to the limited sensor resolution in the systems with P-controller is shown in analogy to [1]. Because of the poor transient response of the P-controlled system the extension to the PD-controller is thereafter taken into consideration. A simple solution for limit cycles avoidance in terms of modification of controller structure is provided. The experimental data confirm the theoretical analysis for the robotics actuators.
I. INTRODUCTION For robots acting in contact with an environment or with humans, a very fast response time for the actuators is required in order to accurately track (or haptically display) forces and positions. For high performance control of such mechanical systems continuous time controller design is not feasible any more. One has to go to the limits provided by the sensor resolution and the computing power. Therefore, the effects of the time discretization, as well as of the sensor quantization have to be taken into account. A controller which is passive in a continuous time representation, might become active due to a time discrete implementation and due to the sensor quantization. This problem is encountered for example for force-feedback and virtual reality systems, and is there a topic of actual research [1]-[7]. These kinds of haptic applications, as well as the use of the DLR Robodrive actuator [12] within the DLR robot joints [11] and for fast direct drive positioning, motivated the present work. The goal is to provide an analysis of the two mentioned discretization effects on the control stability for various controllers. Furthermore, the analysis implicitly provides estimations regarding required sampling time and sensor resolution. Many authors have considered time discretization and sensor quantization effects in their works in the field of haptics. A detailed analysis of the sampling effect has been carried out in the works [4]-[6]. A classical discrete time theory has been used in order to built the haptic system to satisfy stability criterion. Moreover, [6] has also considered a delay effect along with the sampling effect to derive a condition for the stable haptic system. Energy based approaches have been used in [1]-[3] to derive the condition for the passivity of haptic systems considering the sampling and quantization effects together. It has been shown that the passivity of the haptic systems can be lost not only due to the time discretization (as
1-4244-0912-8/07/$25.00 ©2007 IEEE.
shown already in [4]-[6] and [7]), but also due to the sensor quantization. In the works [1]-[2] a very interesting analysis has been carried out showing how the combined effects of nonidealities limit the performances of the haptic systems, measured as the largest obtainable feedback gain. The haptic system in these works was modelled accounting for the hard nonlinearities of quantization, discretization, and delays in the controller, while considering viscous and Coulomb frictions in the mechanism. It is shown that Coulomb friction plays a big role in preserving the passivity of haptic systems, by cancelling the effects of sensor quantization. The above mentioned effects of the sampling time and the sensor quantization play also a big role on the control performance of the robotics actuators. However, for a Pcontroller, as analyzed in [1]-[3], one cannot obtain a good transient performance. This was the motivation for extending the analysis to a PD-controller in the present work. Furthermore, we propose a simple solution for dealing with the sensor quantization effect, if the Coulomb friction is not too high to prevent itself the limit cycles. These are the main contributions of the paper and can be regarded as an extension and generalization of the results in [1]-[3]. The paper is organized as follows: in section II a nonlinear discrete system model with P-controller is analyzed. The analysis refers to the describing function theory that is used to predict the existence of the limit cycles due to the sensor quantization, as well as to determine their stability. Section III extends the analysis of section II to the case of the system with a PD-controller. Section IV provides a simple solution in order to avoid limit cycles. In section V we show the experimental validation of the analytical results obtained in previous sections. Finally some conclusions are presented in section VI. II. PROBLEM STATEMENT A. System Description In this work we are going to model the robotic actuator as a second order nonlinear discrete system, as depicted in Fig. 1. We consider our system in the regulation problem, therefore Xdes is a step function. Our model consist of a physical inertia J and has an intrinsic viscous friction b. It is sampled with a sampling time T . The output of the system X is measured by the sensor with the resolution 4 and provided to the discrete controller K through the feedback.
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Fig. 1.
Second order nonlinear discrete system
Fig. 3.
Fig. 2.
Structure of general sensor
Z
Second order nonlinear discrete system: simplified model
1 π equan (t) cos (nω t)d(ω t) (4) π −π Z 1 π bn = equan (t) sin (nω t)d(ω t) (5) π −π As our nonlinearity is odd and symmetric about the origin, one has ao = 0. Due to the low-pass property of the linear part in our system we have to consider only the fundamental component equan (t), namely an =
In such systems, a limit cycle behavior may arise due to the sensor quantization. The worst-case happens when a desired value is given in the middle of the sensor resolution tick. For the simplicity of the future analysis of our nonlinear discrete system we approximate the sampler and the zeroorder hold element ZOH by a time delay of half sampling T period e−s 2 1 . Then we can redraw the structure of our model for the worst case, as it is shown in Fig. 2. The obtained new model of the feedforward sensor we name as a worst-case sensor. B. Describing Function Analysis As we have mentioned before in this work we are going to focus on the rise of the limit cycles in the system due to the nonlinearity, in our case due to the sensor quantization. Describing function analysis can be conveniently used to discover the existence of limit cycles and determine their stability. The applicability to limit cycles analysis is due to the fact that the form of the signals in a limit-cycling systems is usually approximately sinusoidal. If there exist limit cycles, then all signals in the system must be periodical. The input to the linear part of the system usually consists of many harmonics, and since the linear element filters out all high harmonics because of its low-pass property, the output of the linear element can be approximated as a sinusoid as well. For the analysis of our system depicted in Fig. 2 we assume the existence of the sinusoidal oscillations with amplitude A at the input of the worst-case sensor: e(t) = Asin(ω t)
(1)
Then the output of the nonlinear element equan (t) can be expanded as Fourier series equan (t) =
∞ ao + ∑ [an cos (nω t) + bn sin (nω t)] 2 n=1
where the Fourier coefficients are determined by Z 1 π equan (t)d(ω t) ao = π −π 1 see
Juergen Ackermann, ”Sampled-Data Control Systems”, p. 11
(2)
(3)
equan (t) ≈ a1 cos (ω t) + b1 sin (ω t) = M sin (ω t + φ )
(6)
where M(A, ω ) =
q a21 + b21 and φ (A, ω ) = arctan ( ab11 ).
The describing function of the nonlinear element is given as the complex ratio of the fundamental component of the nonlinear element by the input sinusoid Me j(ω t+φ ) M 1 = e jφ = (b1 + ja1 ). (7) Ae jω t A A Let us now find the describing function of the sensor using above technique. The general structure of sensor is shown in Fig. 3. where ½ 0≤a≤4 0 ≤ bo f f ≤ 4 N(A, ω ) =
For this type of sensor a1 = 0, because of the odd property of nonlinearity. Then for describing function of the sensor we obtain the following expression Z 1 π b1 = equan (t) sin (ω t)d(ω t) = N(A) = A π A −π Z π 2 4 equan (t) sin (ω t)d(ω t) (8) = πA 0 Substituting instead of equan (t) concrete values we obtain the following expression for the describing function of our general sensor (one has to refer [10] for the complete derivation of this describing function) h i A−a r r 4 ³ a ´2 ³ a + i4 ´2 ´ 4 ³ bo f f 1 − N(A) = +4 ∑ 1− πA A A i=1 (9)
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Fig. 4. Describing function of the worst-case sensor: solid line - graphic of the function N(M), dashed line - graphic of the function representing only first term of (11)
For the worst-case sensor we have a=0 bo f f = 4 2 After substituting these parameters into (9) we obtain the describing function of the worst-case sensor £ ¤ 24 44 N(A) = + π A π A2
A 4
q
∑
A2 − (i4)2
(10)
i=1
We can rewrite the last expression not in terms of ampliA tude A, but in terms of sensor ticks M = 4 . N(M) =
[M] p
2 4 + ∑ π M π M 2 i=1
M 2 − i2 .
(11)
C. Analysis of the limit cycles existence If in our system limit cycles behavior occurs, then the following equation must have a solution (or solutions): 1 N(A)
T K e− j 2 ω jω ( jω J + b)
Let us denote T2 ω = x, then we can rewrite the last equation in the following form bT 2J
(20)
For real mechanical systems the value bT 2J is expected to bT be sufficiently small ( 2J 1): for the values of 2) Solution for big amplitude ( 4 amplitude being bigger than one quantization tick we can approximate the describing function of the sensor with unity or in other words, we have unit feedback, no error. Thus, we obtain just a discrete linear system and the condition for the existence of self-sustained oscillations looks according to [6] like 2b (26) T= K III. E XTENSION TO THE PD- CONTROLLER A. System Description Now we are going to extend our analysis for the nonlinear discrete system with a PD-controller. The analysis is a bit more laborious than in case of P-controller, therefore it will be presented separately. The effect of worst-case sensor quantization becomes apparent in case of the small amplitude of motions. Increasing the amplitude this effect diminishes. According to our simulations and experimental results we expect to obtain
Nonlinear discrete system with PD-controller and relay
limit cycles with amplitude smaller than one sensor tick, therefore we can substitute in our analysis the worst-case sensor with a relay element, as shown in Fig. 6. The next step to analyze existence of the self-sustained oscillations in the system. Unfortunately, compared to the system with only P-controller, where we substituted the sampler and a zero-order-hold element with the delay of half sampling period, thus avoiding dealing explicitly with time discretization (analysis of nonlinear continuous system), the system with PD-controller can’t be easily simplified. Therefore, we have to deal with nonlinear discrete system in a explicit way. For the proof of limit cycles existence we are going to use a similar approach, as in case of Pcontroller, but extended to the class of discrete systems, namely sampled describing function analysis. B. Sampled Describing Function Analysis The limit cycles which are most commonly observed in sampled nonlinear systems have periods which are whole multiples of the sampling period ( ωωs = 1n ) [9]. Moreover, we shall center attention on the even fractions, 12 , 14 , 16 ,..., since these are the limit cycle modes one might expect to see in the very common case in which the linear part of the system includes a pole at the origin, an integration. Due to the laborious calculation of sampled describing function for the relay element, we present here only the derived formula. In case of the whole understanding of the derivation of this formula, one has to refer to [9] or [10]. The sampled describing function of the nonlinear element is the complex ratio of the fundamental component of the sampled output of the nonlinear element by the input sinusoid: N(A, ϕ ) =
M·e j(ω t+ψ ) A·e j(ω t+ϕ )
=
M A
· e j(ψ −ϕ )
(27)
Referring to the type of nonlinear element as relay we derived for the all relevant frequencies the parameters of the describing function v" #2 " n −1 #2 u n −1 ³ ³ 2 2 i´ i´ 24 u t M= · ∑ sin 2π · n + ∑ cos 2π · n + 1 Ts n i=1 i=1 (28) n ³ ´ 2 −1 i " ∑ cos 2π · +1# n i=1 ψ = arctan (29) n −1 ³ 2 i´ ∑ sin 2π · n i=1
ϕ is the lag between the zero crossing of e(t) = Asin(ω t) and the next sampling point that is unknown a priori, evidently it can take any value between 0 and 2nπ .
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As one can see there might be several sampled describing functions depending on the value of n. Let us present here some of them. 1) n = 2 → Limit cycles frequency equals to ω2s The sampled describing function of relay element with limit cycle frequency ω = ω2s is N(A, ϕ ) =
4 j( π −ϕ ) ·e 2 Ts A
(30)
The range of the lag variation is 0
TuD2.1
Influence of sensor quantization on the control performance of robotics actuators Renat Iskakov, Alin Albu-Schaeffer, Manfred Schedl, Gerd Hirzinger and Vitaly Lopota* Institute of Robotics and Mechatronics, German Aerospace Center (DLR), Wessling, Germany E-mails: {Renat.Iskakov, Alin.Albu-Schaeffer, Manfred.Schedl, Gerd.Hirzinger}@dlr.de *S.P. Korolev Rocket and Space Corporation - Energia, Korolev, Moscow Region, Russia E-mail: [email protected]
Abstract— In this paper the effect of sensor quantization on the control performance of robotics actuators in the steadystate condition is considered. First, the existence of a limit cycle mode due to the limited sensor resolution in the systems with P-controller is shown in analogy to [1]. Because of the poor transient response of the P-controlled system the extension to the PD-controller is thereafter taken into consideration. A simple solution for limit cycles avoidance in terms of modification of controller structure is provided. The experimental data confirm the theoretical analysis for the robotics actuators.
I. INTRODUCTION For robots acting in contact with an environment or with humans, a very fast response time for the actuators is required in order to accurately track (or haptically display) forces and positions. For high performance control of such mechanical systems continuous time controller design is not feasible any more. One has to go to the limits provided by the sensor resolution and the computing power. Therefore, the effects of the time discretization, as well as of the sensor quantization have to be taken into account. A controller which is passive in a continuous time representation, might become active due to a time discrete implementation and due to the sensor quantization. This problem is encountered for example for force-feedback and virtual reality systems, and is there a topic of actual research [1]-[7]. These kinds of haptic applications, as well as the use of the DLR Robodrive actuator [12] within the DLR robot joints [11] and for fast direct drive positioning, motivated the present work. The goal is to provide an analysis of the two mentioned discretization effects on the control stability for various controllers. Furthermore, the analysis implicitly provides estimations regarding required sampling time and sensor resolution. Many authors have considered time discretization and sensor quantization effects in their works in the field of haptics. A detailed analysis of the sampling effect has been carried out in the works [4]-[6]. A classical discrete time theory has been used in order to built the haptic system to satisfy stability criterion. Moreover, [6] has also considered a delay effect along with the sampling effect to derive a condition for the stable haptic system. Energy based approaches have been used in [1]-[3] to derive the condition for the passivity of haptic systems considering the sampling and quantization effects together. It has been shown that the passivity of the haptic systems can be lost not only due to the time discretization (as
1-4244-0912-8/07/$25.00 ©2007 IEEE.
shown already in [4]-[6] and [7]), but also due to the sensor quantization. In the works [1]-[2] a very interesting analysis has been carried out showing how the combined effects of nonidealities limit the performances of the haptic systems, measured as the largest obtainable feedback gain. The haptic system in these works was modelled accounting for the hard nonlinearities of quantization, discretization, and delays in the controller, while considering viscous and Coulomb frictions in the mechanism. It is shown that Coulomb friction plays a big role in preserving the passivity of haptic systems, by cancelling the effects of sensor quantization. The above mentioned effects of the sampling time and the sensor quantization play also a big role on the control performance of the robotics actuators. However, for a Pcontroller, as analyzed in [1]-[3], one cannot obtain a good transient performance. This was the motivation for extending the analysis to a PD-controller in the present work. Furthermore, we propose a simple solution for dealing with the sensor quantization effect, if the Coulomb friction is not too high to prevent itself the limit cycles. These are the main contributions of the paper and can be regarded as an extension and generalization of the results in [1]-[3]. The paper is organized as follows: in section II a nonlinear discrete system model with P-controller is analyzed. The analysis refers to the describing function theory that is used to predict the existence of the limit cycles due to the sensor quantization, as well as to determine their stability. Section III extends the analysis of section II to the case of the system with a PD-controller. Section IV provides a simple solution in order to avoid limit cycles. In section V we show the experimental validation of the analytical results obtained in previous sections. Finally some conclusions are presented in section VI. II. PROBLEM STATEMENT A. System Description In this work we are going to model the robotic actuator as a second order nonlinear discrete system, as depicted in Fig. 1. We consider our system in the regulation problem, therefore Xdes is a step function. Our model consist of a physical inertia J and has an intrinsic viscous friction b. It is sampled with a sampling time T . The output of the system X is measured by the sensor with the resolution 4 and provided to the discrete controller K through the feedback.
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Fig. 1.
Second order nonlinear discrete system
Fig. 3.
Fig. 2.
Structure of general sensor
Z
Second order nonlinear discrete system: simplified model
1 π equan (t) cos (nω t)d(ω t) (4) π −π Z 1 π bn = equan (t) sin (nω t)d(ω t) (5) π −π As our nonlinearity is odd and symmetric about the origin, one has ao = 0. Due to the low-pass property of the linear part in our system we have to consider only the fundamental component equan (t), namely an =
In such systems, a limit cycle behavior may arise due to the sensor quantization. The worst-case happens when a desired value is given in the middle of the sensor resolution tick. For the simplicity of the future analysis of our nonlinear discrete system we approximate the sampler and the zeroorder hold element ZOH by a time delay of half sampling T period e−s 2 1 . Then we can redraw the structure of our model for the worst case, as it is shown in Fig. 2. The obtained new model of the feedforward sensor we name as a worst-case sensor. B. Describing Function Analysis As we have mentioned before in this work we are going to focus on the rise of the limit cycles in the system due to the nonlinearity, in our case due to the sensor quantization. Describing function analysis can be conveniently used to discover the existence of limit cycles and determine their stability. The applicability to limit cycles analysis is due to the fact that the form of the signals in a limit-cycling systems is usually approximately sinusoidal. If there exist limit cycles, then all signals in the system must be periodical. The input to the linear part of the system usually consists of many harmonics, and since the linear element filters out all high harmonics because of its low-pass property, the output of the linear element can be approximated as a sinusoid as well. For the analysis of our system depicted in Fig. 2 we assume the existence of the sinusoidal oscillations with amplitude A at the input of the worst-case sensor: e(t) = Asin(ω t)
(1)
Then the output of the nonlinear element equan (t) can be expanded as Fourier series equan (t) =
∞ ao + ∑ [an cos (nω t) + bn sin (nω t)] 2 n=1
where the Fourier coefficients are determined by Z 1 π equan (t)d(ω t) ao = π −π 1 see
Juergen Ackermann, ”Sampled-Data Control Systems”, p. 11
(2)
(3)
equan (t) ≈ a1 cos (ω t) + b1 sin (ω t) = M sin (ω t + φ )
(6)
where M(A, ω ) =
q a21 + b21 and φ (A, ω ) = arctan ( ab11 ).
The describing function of the nonlinear element is given as the complex ratio of the fundamental component of the nonlinear element by the input sinusoid Me j(ω t+φ ) M 1 = e jφ = (b1 + ja1 ). (7) Ae jω t A A Let us now find the describing function of the sensor using above technique. The general structure of sensor is shown in Fig. 3. where ½ 0≤a≤4 0 ≤ bo f f ≤ 4 N(A, ω ) =
For this type of sensor a1 = 0, because of the odd property of nonlinearity. Then for describing function of the sensor we obtain the following expression Z 1 π b1 = equan (t) sin (ω t)d(ω t) = N(A) = A π A −π Z π 2 4 equan (t) sin (ω t)d(ω t) (8) = πA 0 Substituting instead of equan (t) concrete values we obtain the following expression for the describing function of our general sensor (one has to refer [10] for the complete derivation of this describing function) h i A−a r r 4 ³ a ´2 ³ a + i4 ´2 ´ 4 ³ bo f f 1 − N(A) = +4 ∑ 1− πA A A i=1 (9)
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Fig. 4. Describing function of the worst-case sensor: solid line - graphic of the function N(M), dashed line - graphic of the function representing only first term of (11)
For the worst-case sensor we have a=0 bo f f = 4 2 After substituting these parameters into (9) we obtain the describing function of the worst-case sensor £ ¤ 24 44 N(A) = + π A π A2
A 4
q
∑
A2 − (i4)2
(10)
i=1
We can rewrite the last expression not in terms of ampliA tude A, but in terms of sensor ticks M = 4 . N(M) =
[M] p
2 4 + ∑ π M π M 2 i=1
M 2 − i2 .
(11)
C. Analysis of the limit cycles existence If in our system limit cycles behavior occurs, then the following equation must have a solution (or solutions): 1 N(A)
T K e− j 2 ω jω ( jω J + b)
Let us denote T2 ω = x, then we can rewrite the last equation in the following form bT 2J
(20)
For real mechanical systems the value bT 2J is expected to bT be sufficiently small ( 2J 1): for the values of 2) Solution for big amplitude ( 4 amplitude being bigger than one quantization tick we can approximate the describing function of the sensor with unity or in other words, we have unit feedback, no error. Thus, we obtain just a discrete linear system and the condition for the existence of self-sustained oscillations looks according to [6] like 2b (26) T= K III. E XTENSION TO THE PD- CONTROLLER A. System Description Now we are going to extend our analysis for the nonlinear discrete system with a PD-controller. The analysis is a bit more laborious than in case of P-controller, therefore it will be presented separately. The effect of worst-case sensor quantization becomes apparent in case of the small amplitude of motions. Increasing the amplitude this effect diminishes. According to our simulations and experimental results we expect to obtain
Nonlinear discrete system with PD-controller and relay
limit cycles with amplitude smaller than one sensor tick, therefore we can substitute in our analysis the worst-case sensor with a relay element, as shown in Fig. 6. The next step to analyze existence of the self-sustained oscillations in the system. Unfortunately, compared to the system with only P-controller, where we substituted the sampler and a zero-order-hold element with the delay of half sampling period, thus avoiding dealing explicitly with time discretization (analysis of nonlinear continuous system), the system with PD-controller can’t be easily simplified. Therefore, we have to deal with nonlinear discrete system in a explicit way. For the proof of limit cycles existence we are going to use a similar approach, as in case of Pcontroller, but extended to the class of discrete systems, namely sampled describing function analysis. B. Sampled Describing Function Analysis The limit cycles which are most commonly observed in sampled nonlinear systems have periods which are whole multiples of the sampling period ( ωωs = 1n ) [9]. Moreover, we shall center attention on the even fractions, 12 , 14 , 16 ,..., since these are the limit cycle modes one might expect to see in the very common case in which the linear part of the system includes a pole at the origin, an integration. Due to the laborious calculation of sampled describing function for the relay element, we present here only the derived formula. In case of the whole understanding of the derivation of this formula, one has to refer to [9] or [10]. The sampled describing function of the nonlinear element is the complex ratio of the fundamental component of the sampled output of the nonlinear element by the input sinusoid: N(A, ϕ ) =
M·e j(ω t+ψ ) A·e j(ω t+ϕ )
=
M A
· e j(ψ −ϕ )
(27)
Referring to the type of nonlinear element as relay we derived for the all relevant frequencies the parameters of the describing function v" #2 " n −1 #2 u n −1 ³ ³ 2 2 i´ i´ 24 u t M= · ∑ sin 2π · n + ∑ cos 2π · n + 1 Ts n i=1 i=1 (28) n ³ ´ 2 −1 i " ∑ cos 2π · +1# n i=1 ψ = arctan (29) n −1 ³ 2 i´ ∑ sin 2π · n i=1
ϕ is the lag between the zero crossing of e(t) = Asin(ω t) and the next sampling point that is unknown a priori, evidently it can take any value between 0 and 2nπ .
1088
As one can see there might be several sampled describing functions depending on the value of n. Let us present here some of them. 1) n = 2 → Limit cycles frequency equals to ω2s The sampled describing function of relay element with limit cycle frequency ω = ω2s is N(A, ϕ ) =
4 j( π −ϕ ) ·e 2 Ts A
(30)
The range of the lag variation is 0
