Simple tracking controller for unicycle-type mobile robots with velocity and torque constraints

Introduction

Wheeled mobile robots are subject to input constraints due to limited actuator performance. In the literature, input constraints have been taken into consideration for wheeled mobile robots in path planning (Ruiz et al., 2013), navigation (Loizou and Kyriakopoulos, 2008), flocking (Gu and Wang, 2009), formation (Consolini et al., 2008), stabilization (Wang, 2008) and tracking control. In particular, tracking control attracts much attention since it is a fundamental problem in the control community. In the absence of input constraints, many tracking controllers have been proposed using techniques such as sliding mode control (Chwa, 2004), fuzzy logic control (Das and Kar, 2006; Hou et al., 2009), feedback linearization (Chwa, 2010; Kim and Oh, 1999), etc. Given the input constraints, the normalization technique (Jiang et al., 2001), backstepping method (Lee et al., 2001), control Lyapunov function approach (Ren et al., 2005) and model predictive control strategy (Gu and Hu, 2006) have been employed to design velocity-constrained tracking controllers based on the kinematic models of mobile robots.

Considering the fact that the actuators cannot track velocity commands changing fast, we should take into consideration the dynamics of wheeled mobile robots, of which the control torques are bounded. In Chen et al. (2009), a moving horizon H ∞ tracking scheme was proposed under torque saturation and external disturbances. In Do (2013), a one-step ahead backstepping method was developed to design the global path tracking controller under torque constraints. In Huang et al. (2013), an adaptive controller was designed in the presence of torque saturation, parameter uncertainties and external disturbances. In Chen et al. (2012), the finite-time control technique was used to design the torque-constrained tracking controller, guaranteeing local stability. In Do (2013), Huang et al. (2013) and Chen et al. (2012), the estimated bounds of control torques are with complicated forms in terms of design parameters and, consequently, the designed controllers are of conservativeness, resulting in loss of robot manoeuvrability.

Practically, both torques and velocities of the wheeled mobile robots should be restricted for safety purposes and slippage avoidance. This observation leads us to study the tracking control problem of unicycle-type mobile robots in the presence of velocity and torque constraints. In this paper, two first-order filters are designed to produce bounded and uniformly continuous feedback signals for the virtual kinematic tracking controller, guaranteeing tracking stability. Then, a simple dynamic controller is proposed to ensure that the robot velocities can converge to the virtual velocity commands in finite time. A set of four inequality constraints on the control design parameters is derived to ensure the compliance with the velocity and torque constraints. The introduction of filters facilitates the bound estimations of the velocities and torque inputs, which are required to prove the compliance with velocity and torque constraints. Moreover, the special form of filters leads to the inequality constraints on the control design parameters being linear, so that these parameters can be readily computed.

To summarize, the strong points in this paper compared to existing work are threefold. Firstly, in addition to torque constraints, the velocity constraints are explicitly considered, which is more practical to ensure safety and avoid slippage. Secondly, the control design parameters can be easily computed for given velocity and torque constraints. Thirdly, the proposed controller is of less conservativeness so that the robot manoeuvrability can be fully exploited.

The rest of this paper is structured as follows. Section 2 states the tracking control problem of unicycle-type mobile robots subject to velocity and torque constraints. Section 3 presents the main results of controller design. Simulation and experimental results are provided in Section 4, and concluding remarks are given in Section 5.

Problem statement

The unicycle-type mobile robot is of the kinematics

x · = v cos θ , y · = v sin θ , θ · = w

(1)

where ( x , y ) is the robot position, θ is the robot orientation, v is the linear velocity and w is the angular velocity (see Figure 1). The robot dynamics is

v · = τ 1 , w · = τ 2

(2)

where τ 1 and τ 2 are control torques. In practice, the mobile robot is subject to input constraints, that is:

| τ 1 | ≤ α v , | τ 2 | ≤ α w

(3)

where α v and α w are torque bounds. In addition, for safety purposes and slippage avoidance, the robot velocities are restricted, that is:

| v | ≤ λ v , | w | ≤ λ w

(4)

where λ v and λ w are velocity bounds.

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Figure 1.

Trajectory tracking configuration of a unicycle-type mobile robot.

Assumption 1: The reference trajectory for the mobile robot satisfies

x · r = v r cos θ r , y · r = v r sin θ r , θ · r = w r

(5)

with the constraints

0 < v r ≤ γ v < λ v , | w r | ≤ γ w < λ w , | v · r | ≤ μ v < α v , | w · r | ≤ μ w < α w

(6)

where ( x r , y r , θ r , v r , w r ) are the desired values for ( x , y , θ , v , w ) , and γ v , γ w , μ v , μ w are positive constants.

Remark 1: The constraints (6) on the reference trajectory are reasonable, because it is impossible for the mobile robot to track trajectories beyond its manoeuvrability.

We aim to design the control inputs τ 1 and τ 2 in the presence of the torque constraints (3) and the velocity constraints (4) such that the mobile robot will move on the reference trajectory defined by Assumption 1.

Main results

In view of (1) and (5), defining the tracking errors (see Figure 1)

[ x e y e θ e ] = [ cos θ sin θ 0 − sin θ cos θ 0 0 0 1 ] [ x r − x y r − y θ r − θ ]

(7)

yields the tracking error system

x · e = v r cos θ e − v + w y e y · e = v r sin θ e − w x e θ · e = w r − w

(8)

Design the virtual controller

v d = v r + ϕ 1 w d = w r + k v r y e cos θ e 2 − x e sin θ e 2 σ 2 + x e 2 + y e 2 + ϕ 2

(9)

where σ is a positive constant, k is a positive feedback parameter and ϕ 1 , ϕ 2 are feedback functions generated from the first-order filters:

ϕ · 1 = b 1 2 ( − ϕ 1 a 1 + x e σ 2 + x e 2 + y e 2 ) , ϕ 1 ( 0 ) = 0 ϕ · 2 = b 2 2 ( − ϕ 2 a 2 + sin θ e 2 ) , ϕ 2 ( 0 ) = 0

(10)

where a 1 , a 2 , b 1 and b 2 are positive filter parameters. Defining the velocity errors

v e = v − v d , w e = w − w d

(11)

we design the torque controller

τ 1 = − ( α v sign ( v e ) + v · d ) sat ( | v e c 1 | p 1 ) + v · d τ 2 = − ( α w sign ( w e ) + w · d ) sat ( | w e c 2 | p 2 ) + w · d

(12)

where c 1 , c 2 , p 1 and p 2 are positive parameters, 0 < p 1 , p 2 < 1 and

sign ( z ) = { | z | / z z ≠ 0 0 z = 0 sat ( z ) = min { | z | , 1 }

(13)

Lemma 1 : (Khalil, 2002) If f : [ 0 , ∞ ) → R is uniformly continuous and lim t → ∞ ∫ 0 t f ( τ ) d τ exists and is finite, then lim t → ∞ f ( t ) = 0 .

Lemma 2: (Jiang and Nijmeijer, 1997) If a bounded scalar function z ( t ) satisfies z → 0 as t → ∞ and z . = − z + g ( t ) , where g ( t ) is bounded and uniformly continuous, then z · → 0 and g ( t ) → 0 as t → ∞ .

Lemma 3: (Bhat and Bernstein, 2000) If a non-negative scalar function V ( t ) satisfies V · + η V ρ ≤ 0 with the constants η > 0 and 0 < ρ < 1 , then there is a time point T 0 , T 0 ≤ V ( 0 ) 1 − ρ η ( 1 − ρ ) , that V ( t ) ≡ 0 for t ≥ T 0 .

Theorem 1: Apply the controller (12) to the mobile robot system (1) and (2) under Assumption 1. If | v ( 0 ) | ≤ λ v , | w ( 0 ) | ≤ λ w , and

γ v + a 1 + c 1 ≤ λ v

(14a)

γ w + k γ v + a 2 + c 2 ≤ λ w

(14b)

μ v + b 1 < α v

(14c)

μ w + k ( μ v + γ v ( γ v + λ v ) / σ + γ v ( γ w + λ w ) / 2 ) + b 2 < α w

(14d)

then the torque constraints (3) and the velocity constraints (4) are satisfied, x e and y e will converge to zero, and θ e will converge to 2 i π , i ∈ Z .

Proof. It is obtained from (10) that

| ϕ 1 | ≤ a 1 , | ϕ 2 | ≤ a 2 , | ϕ · 1 | ≤ b 1 , | ϕ · 2 | ≤ b 2 .

(15)

Then we have from (6), (9), (14a) and (14b) that

| v d | ≤ γ v + a 1 ≤ λ v − c 1 | w d | ≤ γ w + k γ v + a 2 ≤ λ w − c 2 .

(16)

According to (8) and (9)

v · d = v · r + ϕ · 1 w · d = w · r + k v · r y e cos θ e 2 − x e sin θ e 2 σ 2 + x e 2 + y e 2 + k v r d ( y e cos θ e 2 − x e sin θ e 2 σ 2 + x e 2 + y e 2 ) / d t + ϕ · 2

(17)

with

d ( y e cos θ e 2 − x e sin θ e 2 σ 2 + x e 2 + y e 2 ) / d t = v r ( σ 2 sin θ e 2 + ( x e sin θ e − y e cos θ e ) ( x e cos θ e 2 + y e sin θ e 2 ) ) ( σ 2 + x e 2 + y e 2 ) 3 / 2 + v ( σ 2 sin θ e 2 + y e ( x e cos θ e 2 + y e sin θ e 2 ) ) ( σ 2 + x e 2 + y e 2 ) 3 / 2 − ( w r + w ) ( x e cos θ e 2 + y e sin θ e 2 ) 2 σ 2 + x e 2 + y e 2 .

(18)

In view of (6) and (18)

| d ( y e cos θ e 2 − x e sin θ e 2 σ 2 + x e 2 + y e 2 ) / d t | ≤ | v r | σ 2 + x e 2 + y e 2 ( σ 2 + x e 2 + y e 2 ) 3 / 2 + | v | σ 2 + | y e | x e 2 + y e 2 ( σ 2 + x e 2 + y e 2 ) 3 / 2 + | w r + w 2 | x e 2 + y e 2 σ 2 + x e 2 + y e 2 ≤ γ v + | v | σ + γ w + | w | 2 .

(19)

Thus we have from (6), (15) and (17) that

| v · d | ≤ μ v + b 1 | w · d | ≤ μ w + k ( μ v + γ v ( γ v + | v | ) / σ + γ v ( γ w + | w | ) / 2 ) + b 2 .

(20)

Now, we prove that the torque constraints (3) and the velocity constraints (4) are satisfied. Assume that | v ( 0 ) | ≤ λ v and | w ( 0 ) | ≤ λ w . Since | v d | ≤ λ v − c 1 , it is obtained from (12) that v · = − α v for v = λ v and v · = α v for v = − λ v , and consequently | v ( t ) | ≤ λ v . Similarly, we can get | w ( t ) | ≤ λ w . Then from (14d) and (20), we get

α w − | w · d | ≥ d w > 0

(21)

where

d w = α w − ( μ w + k ( μ v + γ v ( γ v + λ v ) / σ + γ v ( γ w + λ w ) / 2 ) + b 2 ) .

(22)

In view of (12) and (21), | τ 2 | = | w · d | < α w for w e = 0 , and for w e > 0

− α w ≤ τ 2 = − ( α w + w · d ) sat ( | w e c 2 | p 2 ) + w · d < w · d < α w

(23)

as well as for w e < 0

− α w < w · d < τ 2 = ( α w − w · d ) sat ( | w e c 2 | p 2 ) + w · d ≤ α w

(24)

since 0 < sat ( | w e c 2 | p 2 ) ≤ 1 for w e ≠ 0 , α w + w · d ≥ α w − | w · d | > 0 and α w − w · d ≥ α w − | w · d | > 0 . Thus, | τ 2 | ≤ α w . In view of (14c) and (20), we have

α v − | v · d | ≥ d v > 0

(25)

where

d v = α v − ( μ v + b 1 )

(26)

Similar to the analysis of τ 2 , we get from (12) and (25) that | τ 1 | ≤ α v .

Next, we analyse the system stability. Defining the Lyapunov function V w = w e 2 / 2 , we have from (11), (12) and (21) that

V · w = − ( α w + w · d sign ( w e ) ) | w e | sat ( | w e c 2 | p 2 ) ≤ − d w | w e | sat ( | w e c 2 | p 2 )

(27)

If | w e | > c 2 , we have d ( | w e | ) / d t ≤ − d w since V · w = | w e | d ( | w e | ) / d t . This means | w e | ≤ c 2 for t ≥ 2 λ w d w because of | w e | ≤ | w | + | w d | < 2 λ w . Once | w e | ≤ c 2 , V · w ≤ − 2 p 2 + 1 2 d w c 2 − p 2 V w p 2 + 1 2 . According to Lemma 3, w e ≡ 0 for t ≥ T 1 = 2 λ w d w + c 2 d w ( 1 − p 2 ) . Similarly, we can get that v e ≡ 0 for t ≥ T 2 = 2 λ v d v + c 1 d v ( 1 − p 1 ) . Choose the Lyapunov function

V d ( t ) = σ 2 + x e 2 + y e 2 − σ + 4 k ( 1 − cos θ e 2 ) + ϕ 1 2 b 1 + 2 ϕ 2 2 k b 2

(28)

For t ≥ T 3 = max { T 1 , T 2 } , since v e ≡ 0 and w e ≡ 0 , we have from (8)–(11) that

V · d = x e x · e + y e y · e σ 2 + x e 2 + y e 2 + 2 k θ · e sin θ e 2 + 2 ϕ 1 ϕ · 1 b 1 + 4 ϕ 2 ϕ · 2 k b 2 = − ϕ 1 2 a 1 − 2 ϕ 2 2 k a 2 ≤ 0

(29)

which indicates that V · d is uniformly continuous due to the uniform continuity of ϕ 1 and ϕ 2 (see (15)). As V d ( t ) is non-increasing and lower bounded, ∫ 0 ∞ V · d d t = V d ( ∞ ) − V d ( 0 ) is finite. Applying Lemma 1 ensures that V · d → 0 as t → ∞ ; that is, ϕ 1 and ϕ 2 converge to zero. In view of (28) and (29), x e and y e are bounded. It follows from (6), (8) and (16) that x · e , y · e and θ · e are also bounded. Thus, x e σ 2 + x e 2 + y e 2 , y e σ 2 + x e 2 + y e 2 , sin θ e 2 and cos θ e 2 are uniformly continuous. In view of (10), it is concluded from Lemma 2 that x e and sin θ e 2 will converge to zero. Notice that sin θ e 2 → 0 indicates θ e → 2 i π , i ∈ Z .

Since w ≡ w d for t > T 3 , we get from (8) and (9) that

θ · e = − k v r y e cos θ e 2 − x e sin θ e 2 σ 2 + x e 2 + y e 2 − ϕ 2

(30)

which is uniformly continuous by the above analysis and the uniform continuity of v r . Since sin θ e 2 → 0 as t → ∞ , ∫ 0 ∞ θ · e = θ e ( ∞ ) − θ e ( 0 ) is finite. Applying Lemma 1 gives that θ · e → 0 . Since ϕ 2 , x e and sin θ e 2 converge to zero and v r > 0 , y e → 0 as t → ∞ .

Remark 2: It is observed that the set of inequality constraints (14), guaranteeing the compliance with (3) and (4), is actually derived from the bound estimations of v d , w d , v · d and w · d . The introduction of filters (10) for ϕ 1 and ϕ 2 facilitates the bound estimations, since they produce amplitude and rate-bounded signals and the bounds of ϕ 1 , ϕ 2 , ϕ · 1 and ϕ · 2 are easily estimated due to the special form of filters. In addition, the filters in the presented form result in that the inequality constraints on the control design parameters are linear, so that these parameters can be easily computed for any specified velocity and torque bounds.

Remark 3: The introduction of the filters (10) relaxes the inequality constraints imposed on the control design parameters compared to the counterpart (11) in Chen et al. (2012), although we further consider the velocity constraints. The number of inequality constraints on the design parameters in Chen et al.’s work is 14, and some of the inequality constraints are nonlinear in terms of the design parameters. In contrast, the control design parameters in our work are only constrained by four linear inequality constraints. Thus, the control design parameters can be easily calculated. The relaxation of inequality constraints on the control design parameters ensures the better performance of our tracking controller than that of Chen et al. (2012), which is confirmed by the following simulation results.

Remark 4: The control torques τ 1 and τ 2 generated from (12) are continuous. The maximum torques are produced when v e and w e are large. In other words, the robot manoeuvrability can be fully utilized to accelerate the tracking convergence.

Simulation and experimental results

Simulation results

We compare the performance between the controller used in the simulation of Chen et al. (2012) (referred to as Controller 1) and our controller (referred to as Controller 2). The simulation configuration is the same as that in Chen et al. (2012); that is, the reference trajectory is a circle with v r = 1 . 5 m / s , w r = 1 . 2 rad / s and ( x r , y r , θ r ) | t = 0 = ( 0 , 0 , 0 ) , the initial robot state is ( x , y , θ , v , w ) | t = 0 = ( 0 . 2 m , − 0 . 3 m , 0 , 1 . 0 m / s , 1 . 2 rad / s ) , and γ v = 1 . 5 , γ w = 1 . 2 , μ v = 2 . 0 , μ w = 1 . 8 . The design parameters of Controller 2 are k = 0 . 1 , σ = 0 . 5 , a 1 = 1 . 0 , a 2 = 0 . 5 , b 1 = 4 . 0 , b 2 = 1 . 0 , c 1 = c 2 = 0 . 1 and p 1 = p 2 = 0 . 8 . The velocity bounds are λ v = 2 . 6 and λ w = 2 . 0 . The torque bounds are α v = 6 . 1 and α w = 4 . 6 , which are less than α v = 12 . 5 and α w = 12 . 2 for Controller 1 in Chen et al. (2012). It can be calculated that (14a)–(14d) are satisfied.

Figure 2 shows the tracking trajectories of the first 10 seconds. It is observed that only Controller 2 drives the robot on the reference trajectory by t = 10 s . Figure 3 demonstrates that Controller 2 results in much faster convergence rate of tracking errors than Controller 1. It takes about 10 seconds for Controller 2 to track the reference trajectory, while small tracking errors still exist at t = 40 s for Controller 1. Figure 4 shows that the velocity constraints and the torque constraints are satisfied. The torque inputs and the velocities for Controller 2 converge to the steady values faster compared to Controller 1. Since the constraints (14a)–(14d) are less conservative than the counterpart (11) in Chen et al. (2012), Controller 2 is of better performance than Controller 1 although the maximum control torques of Controller 2 are less than that of Controller 1.

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Figure 2.

Simulation results: reference and tracking trajectories for t ∈ [ 0 , 10 s ] .

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Figure 3.

Simulation results: tracking errors.

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Figure 4.

Simulation results: velocities and torque inputs.

Experimental results

The experimental system is a Pioneer 3-DX differential-drive mobile robot manufactured by Adept Inc. Optical encoders are installed on the driving motor axes. The encoder reading is used to measure the robot velocities. The robot position and orientation are measured by fusing the encoder reading and the gyro measurement. Figure 5 shows the tracking control scheme. The reference trajectory generator produces the realtime reference pose for the mobile robot. The differences between the measured and reference poses are passed to the tracking controller after error transformation. The tracking controller uses the information of tracking errors, reference velocities and reference accelerations to generate torque control commands. Since the Pioneer 3-DX robot only provides the velocity control interface, the velocity command generator is employed to produce the velocity commands v c and w c as

v c = v + T τ 1 , w c = w + T τ 2

(31)

where T = 0 . 05 s is the control cycle. All the blocks in Figure 5 are programmed in C++ language and run on the onboard computer of the mobile robot.

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Figure 5.

Tracking control scheme of the Pioneer 3-DX mobile robot.

In the experiment, the torque and velocity bounds of the mobile robot are set as α v = 0 . 6 , α w = 1 . 0 and λ v = λ w = 1 . 0 . The reference trajectory is

x r = 3 sin ( 2 π t / 50 ) , y r = 2 sin ( 4 π t / 50 )

(32)

of which γ v = 0 . 63 , γ w = 0 . 57 , μ v = 0 . 1 and μ w = 0 . 18 . The parameters are chosen as k = 0 . 1 , σ = 0 . 4 , a 1 = 0 . 25 , a 2 = 0 . 3 , b 1 = 0 . 45 , b 2 = 0 . 35 , c 1 = c 2 = 0 . 06 and p 1 = p 2 = 0 . 7 to satisfy (14a)–(14d). The initial robot state is ( x , y , θ , v , w ) | t = 0 = ( − 1 . 2 m , 1 . 0 m , 0 , 0 , 0 ) .

It is observed from Figure 6 that the mobile robot gradually converges to the reference trajectory and finally travels along it. The tracking errors presented in Figure 7 approach small neighbourhoods of zero. Figure 8 shows that the torques and velocities are restricted in preset bounds. At the initial four seconds, τ 1 is maintained at the maximum while v only grows up to 0 . 6 m / s , which is because the initial response of the driving motors takes seconds. The residual tracking errors in Figure 7 and the chattering of torque inputs in Figure 8 result from the imperfect command execution of driving motors, measurement noises and external disturbances.

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Figure 6.

Experimental results: reference and tracking trajectories.

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Figure 7.

Experimental results: tracking errors.

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Figure 8.

Experimental results: velocities and torques.

Conclusions

We addressed the constrained tracking control problem of unicycle-type mobile robots. In addition to the torque constraints, the velocity constraints were explicitly taken into consideration. A simple tracking controller incorporating two first-order filters was proposed. The use of filters greatly simplified the bound estimations of velocities and torques, and yielded the relaxed inequality constraints on the control design parameters compared to existing work. The simulation comparison showed better performance and less conservativeness of our controller, and the experimental results demonstrated the practical use of the proposed controller.