Control of a Quadrotor and a Ground Vehicle Manipulating an Object\thanksreffootnoteinfo

Control of a Quadrotor and a Ground Vehicle Manipulating an Object\thanksreffootnoteinfo

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Abstract

This paper focuses on the control of a cooperative system composed of an Unmanned Aerial Vehicle (UAV) and an Unmanned Ground Vehicle (UGV) manipulating an object. The two units are subject to input saturations and collaborate to move the object to a desired pose characterized by its position and inclination. The dynamics are derived using Euler-Lagrange method. A pre-stabilizing control law is proposed where the UGV is tasked to deploy the object to a certain position whereas the UAV adjusts its inclination. In particular, a proportional-derivative control law is proposed for the UGV, and a cascade control approach is used for the UAV, where the inner loop controls the attitude of the UAV and the outer loop stabilizes the inclination of the object. Then, we prove the stability of the points of equilibrium using small gain arguments. To ensure constraints satisfaction at all times, a reference governor unit is added to the pre-stabilizing control scheme. Finally, numerical results combined with experimental results are provided to validate the effectiveness of the proposed control scheme in practice.

ULB]Tam W. Nguyen, ULB]Laurent Catoire, ULB]Emanuele Garone

Université libre de Bruxelles, Av. F. Roosevelt 50, CP 165/55, 1050, Brussels, Belgium 


Key words:  Autonomous vehicles; Manipulation tasks; Robotic manipulators; Nonlinear control systems; Constraint satisfaction problems.

 

11footnotetext: This work is supported by FRIA ”COPTERS”, Ref. F-3-5-5/FRIA/FC-12687. The preliminary results of this paper were presented at the IEEE American Control Conference (ACC), 2016 (see [7]).

1 Introduction

Unmanned Aerial Vehicles (UAVs) have been so far used for remote sensing to perform e.g., aerial photography [17], monitoring [12], and agriculture [1]. In very recent years, the study of their interactions with the environment has attracted the interest of researchers, giving rise to the new field of Aerial Robotics [3, 6]. Research on this topic aims at extending the use of UAVs to more complex missions where UAVs physically interact with the environment and with other robots. Most works on aerial manipulation deal with the transportation of objects through single and multiple UAVs, including works on grasping [11], and cooperative transportation [4, 10]. As aerial manipulation became more popular and more sophisticated, it has also triggered other research initiatives such as making UAVs capable to collaborate with Unmanned Ground Vehicles (UGVs). Early works on the physical interaction between UAVs and UGVs include the pulling of a cart through one or two quadrotors [15], the cooperative pose stabilization of a UAV through a team of ground robots [5] and the modeling and control [8, 18] of tethered UAVs. To the best of the authors’ knowledge, the study of the manipulation of objects using a team of aerial and ground vehicles is still in its early phase of development. Very few studies on the subject exist in the literature. For example, our preliminary paper [7] proposes a first control law for a UAV and a UGV manipulating an object subject to actuator saturations. In [16], the authors propose a controller for tracking. This paper makes use of model inversion techniques, which can be problematic in the presence of e.g., model uncertainties and disturbances. In this paper, we propose to control the cooperative system using a control law (based on proportional-derivative law) which is inherently robust as a highly accurate model is not required to design the controller. This paper substantially extends the preliminary results proposed by the same authors in [7]. The results of [7] were derived under the assumption that the sum of the masses of the UAV and of the object were negligible with respect to the UGV mass, resulting in a non Euler-Lagrange system. In this paper, we discard this limitative assumption, introducing new coupling dynamics and resulting in a more complete and mechanically correct dynamics (directly derived from Euler-Lagrange). In this context, the stability proofs of [7] become inapplicable and a new input-to-state (ISS) Lyapunov is introduced to prove stability. The stability results of this paper are new and represent the main contribution of this paper. Furthermore, this paper provides more extensive simulations combined with new experimental results to validate the model and the effectiveness of the proposed control scheme.

This paper is organized as follows. First, the complete dynamics of the system are derived using Euler-Lagrange methods. Then, the attainable configurations of equilibrium are computed considering the saturations of the actuators. Afterwards, a control scheme is proposed where the stability of the points of equilibrium of the system is proved using small gain arguments and strict Lyapunov functions. To ensure constraints satisfaction at all times, the control scheme is augmented with the Reference Governor (RG). Finally, numerical and experimental results are compared and discussed.

2 Problem Statement

Consider the planar model of a quadrotor UAV\@footnotemark\@footnotetextIn this paper, we consider a quadrotor UAV with a particular fixed yaw angle so that the quadrotor can be considered as a birotor in 2D. Note that the thrust and the torque of the birotor in 2D can be mapped to the quadrotor by suitably distributing the forces to the four propellers. and a UGV manipulating a rigid body (object) as depicted in Fig. 1. We assume that the center of mass of the UAV coincides with the joint position where it is attached to.

Fig. 1: Model of a UAV and a UGV manipulating an object.

The UAV has mass and moment of inertia . The UGV has mass and the object has mass , moment of inertia , and length . The center of mass of the object is positioned at distance from the UGV. Let the position of the object (attached to the UGV) , the inclination of the object , and the attitude of the UAV be the generalized coordinates of the system. All the angles are defined counter-clockwise with respect to the horizon, and each body is subject to the gravity acceleration . The UAV propellers generate the total thrust and the resultant torque . The UGV motors produce the force . The signs of the inputs , , and are defined positive with respect to the oriented vectors depicted in Fig. 1. The actuators are saturated as

(1)

where , , and .

The equations of motion of the system are derived from Euler-Lagrange. Assuming friction forces negligible, we can derive the equations of motion as

(2a)
(2b)
(2c)

where is the relative angle between the object and the UAV, is the total mass of the system, represents the apparent mass of the UAV and the object, and is the moment of inertia of the system divided by . Furthermore, we define as the feed-forward control input where is feed-forwarded to , and as the normalized control input.

It is worth noting that system (2a)-(2b) can be represented as an open-chain robotic manipulator since it can be rewritten as

(3)

where , , , , and are the coordinates, the inertia, the Coriolis, the gravity, and the external force matrices, respectively. As detailed in Appendix A, system (2) enjoys the basic properties of open-chain manipulators, i.e., the inertia matrix is positive definite, and is skew-symmetric.

The objective of this paper is to stabilize the pose (i.e., position and orientation) of the object to the desired position and the desired angle by means of the cooperation of the UAV and the UGV. Prior to designing the controllers of each unit, we will first analyze the attainable configurations of equilibrium in the presence of input saturations.

3 Attainable Configurations of Equilibrium

In this section, the attainable configurations of equilibrium and the associated steady-state input vector are computed taking into account constraints (2). Setting all the time derivatives of (2) to zero, it follows that the configurations of equilibrium must satisfy the system of equations

(4a)
(4b)
(4c)

Clearly, Eq. (4c) gives as the only attainable input associated to an equilibrium. Moreover, note that any position is an attainable point of equilibrium since does not appear in (3). Regarding (4a), since we assumed that , the force at equilibrium always exists for any and . For what concerns Eq. (4b), it is possible to compute the maximum value of using the fact that . In particular, the maximum value is reached when and . Accordingly, there are two possible cases:

  • If , then any is an attainable angle of equilibrium;

  • If , then the attainable angles of equilibrium are restricted to the interval , where the boundaries are and .

Finally note that, for a given steady-state angle , the attainable equilibria for the attitude are restricted to the interval . The boundaries of this interval can be computed solving (4b) by substituting . Doing so, we obtain and , where the positive sign is taken if , and the negative one if .

4 Control Scheme

Fig. 2: Proposed control scheme.

The proposed control scheme consists of two separate control units that are controlling the UAV and the UGV (see Fig. 2). The UGV controller generates the control input so as to make the object position asymptotically tend to . The UAV controller uses a cascade control approach, where the inner loop is tasked with the control of the UAV attitude , whereas the outer loop is tasked to control the inclination of the transported object. For constraints satisfaction, an RG unit is added to the scheme. Whenever necessary, the RG modifies the desired references and to and , respectively, to ensure that constraints are satisfied at all times. In the following subsections, the control laws of the UGV and the UAV are detailed.

4.1 UGV Control Law

The objective of the UGV control law is to steer the object to the desired position . For this purpose, the following proportional-derivative (PD) control law is proposed:

(5)

where is the position error, and are the control gains to be tuned.

4.2 UAV Control Law

The UAV control law uses a cascade control approach, where the inner loop controls the UAV attitude and the outer loop controls the inclination of the object.

4.2.1 Inner Loop

To control the UAV attitude, a PD control law is chosen:

(6)

where are control parameters to be tuned, is the attitude error and the desired UAV attitude.

4.2.2 Outer Loop

We define as the desired relative attitude of the UAV, and the tangential force produced by the UAV on the object as

(7)

For the moment, assume that we can use as a new control input to stabilize the inclination of the object ; the proposed control law is a PD with gravity compensation

(8)

where is the object inclination error, and are the parameters to be tuned so that and . It remains to construct and that produce the desired tangential force (4.2.2). In line of principle, Eq. (4.2.2) admits an infinite number of solutions for and . However, in this paper, we propose the following continuous mapping\@footnotemark\@footnotetextWe define the saturation function as

(9)

where are parameters to be chosen such that thrust constraints are satisfied at steady-state. The main advantage of (4.2.2) is that this mapping always guarantees the positiveness of . Indeed, both functions and are odd and monotonically increasing with respect to the variable . Hence, rewriting (4.2.2) as

(10)

the resulting is always positive since the quotient of two odd and monotonically increasing functions is always positive. Another relevant property of (4.2.2) is that does not present any singularities since for any not equal to zero, (4.2.2) is always determined, and . Finally, the mapping (4.2.2) presents the interesting advantage that, if the parameter is properly chosen, we can prove that it is possible to freely choose the parameter (becoming then a tuning parameter) which ensures constraints satisfaction at any attainable configuration of equilibrium. In particular, if we choose so as to satisfy saturations (2) when , it follows from (4.2.2) that . As a result, following from (4.2.2), we obtain

(11)

As clarified in the following Lemma, with fixed as in (4.2.2), steady-state constraints are always ensured for any and, as a consequence, can be freely chosen as a tuning parameter.

Lemma 1

For any , the mapping (4.2.2) with satisfying (4.2.2) ensures at equilibrium.

PROOF. Consider first the particular case where . In view of (4.2.2), the control input at equilibrium must satisfy . Defining the minimum relative UAV attitude , and using found in Section 3, we obtain . Then, following from (4b) and defining , to ensure for all points of equilibrium, must satisfy . As a consequence, the inequality must be satisfied for any . Choosing as in (4.2.2), the inequality holds true for any since, if restricted to , is convex and is concave. The same arguments hold true for any , where with , concluding the proof.

5 Stability Properties

This section is dedicated to prove the asymptotic stability of (2c),(2) using the control law (4.1), (4.2.1), (4.2.2), (4.2.2), and (4.2.2) for . In order to do so, we first prove that (2c) controlled by (4.2.1) is ISS with respect to . Furthermore, we prove that the asymptotic gain of the inner loop can be made arbitrarily small by acting on and . Afterwards, (2) controlled by (4.1), (4.2.2), (4.2.2), and (4.2.2) is proved to be ISS with restriction with respect to . This enables to prove that the asymptotic gain of the outer loop exists and is finite. As a consequence, since the overall system is interconnected, it is possible to prove that the points of equilibrium are asymptotically stable using small gain arguments.

To prove that the inner loop is ISS with respect to , we reformulate the inner attitude dynamics of (2c) controlled by (4.2.1) as

(12a)
(12b)

System (5) represents the dynamics of the inner loop, where the states are affected by the exogenous input . The following property can be proved.

Lemma 2

The inner-loop (5) is ISS with respect to for any and . The asymptotic gain between the disturbance and the output is finite and can be made arbitrarily small for sufficiently large and

PROOF. Please refer to [9, Proposition 15].

The next step is to prove that (2) controlled by (4.1), (4.2.2), (4.2.2) and (4.2.2) is ISS with restriction with respect to and that there is a finite gain between and To do so, let us first rewrite the control action (4.1), (4.2.2), (4.2.2) and (4.2.2) so as to isolate the effect of the attitude error . Since , we can use (4.2.2) and (4.2.2) to rewrite the right-hand side of (2b) as

(13)

where is the second component of , and . At this point, using (4.1) and (5), the external force vector can be rewritten as

(14)

where , , , are the state error, the proportional gain, the derivative gain, and the exogenous input matrices (depending on ) affecting the states of (2), respectively. Interestingly enough, it is possible to prove that can be upper-bounded by a saturated linear function of

Lemma 3

The norm of satisfies for any , , and .

PROOF. Using triangular inequality, it is possible to prove that and that . As a result, the property stated by Lemma 3 holds true. For more details, please refer to Appendix B.

The following Lemma proves that (2) controlled by (4.1), (4.2.2), (4.2.2), and (4.2.2) is ISS with restriction with respect to the attitude error and that there exists a finite asymptotic gain between and

Lemma 4

Given the desired position , the desired inclination , and the resulting steady-state attitude resulting from (4.2.2), system (2) controlled by (5) is ISS with restriction (or equivalently ), and with respect to Furthermore the asymptotic gain between the disturbance and the output exists and is finite\@footnotemark\@footnotetextFor a complete characterization of , and , please refer to (C.58), (C.59), and (C.77), respectively, in Appendix C..

PROOF. The proof uses the Lyapunov function [13]

(15)

where , , is a positive scalar satisfying , is the largest eigenvalue of , is the largest eigenvalue of , and is a positive scalar satisfying . It can be shown that the time derivative of can be bounded by

To prove ISS with restriction, it is enough to use the fact that whenever remains outside a ball of radius with restriction and . The details of the proof can be found in Appendix C.

Combining Lemmas 2 and 4, it is possible to prove the asymptotic stability of the points of equilibrium.

Theorem 5

Consider system (2c),(2) controlled by (4.1), (4.2.1), (4.2.2), and (4.2.2). Given the desired position the desired inclination and the resulting steady-state attitude , the point of equilibrium is asymptotically stable for suitably large and .

PROOF. From Lemmas 2 and 4, and are proved to be finite under the assumption and . In this case, we can achieve since can be made arbitrarily small for sufficiently large and . Therefore, the Small Gain Theorem can be applied and, for a suitable set of initial conditions around the point of equilibrium that satisfies and , the closed loop system is asymptotically stable.

Interestingly enough, it is possible to improve this control law by substituting (4.2.2) with\@footnotemark\@footnotetextIn this paper, we define the positive saturation function as .

(16)

The main difference between (4.2.2) and (16) is that, instead of dividing by the desired relative attitude , the new control law (16) divides directly by the actual relative attitude . The following proposition proves that the new control law (16) preserves the previous stability results.

Proposition 6

Consider (2c),(2) controlled by (4.1), (4.2.1), (4.2.2), (4.2.2) and (16). For any , for any , and for the resulting steady-state , the point of equilibrium is asymptotically stable. Furthermore, the control law (16) is equivalent to the control law (4.2.2) with a feed-forward action.

PROOF. The control law (16) is equivalent to a feedforward block that possibly reduces the effect of the attitude error on the outer loop. In particular, the feedforward block modifies the actual attitude error to . It can be shown that , which makes the gain between and smaller than one. As a result, all the stability results of Theorem 5 apply. For more details, please refer to Appendix D.

Note that the stability results presented in this section are local (i.e., valid for any initial condition sufficiently close to the equilibrium point). In the next section, the presented control scheme will be augmented with an RG, which makes the system asymptotically stable for a larger set of initial conditions, e.g. for any steady-state admissible initial condition satisfying , and with zero initial velocities.

6 Constraints Enforcement

In this section, the control law previously studied is augmented with the RG introduced in [2] to avoid constraints violation and, consequently, increase the basin of attraction of the points of equilibrium. Let the desired position and angle references be given, where and . If needed, the RG substitutes the desired set-point with a sequence of applied way-points that ensures that the system trajectories do not violate the constraints. This sequence is computed online by assuming that, at time , the applied reference , if maintained constant, would not make the system violate the constraints. The RG computes (at fixed time intervals) the next applied reference by maximizing the scalar so that, if this reference were to remain constant, the system trajectories would not violate constraints at any future time instant. The optimization of can be performed using bisection [2] and online simulations over a sufficiently long prediction horizon. The convexity of the set of steady-state admissible equilibria ensures that, if is kept constant, the way-point sequence converges to .

7 Simulations

Fig. 3: Numerical simulations. The solid lines represent the states of the system, the red dash-dot line represents , and the dashed lines represent the applied references.

Consider a UAV of mass and inertia cooperating with a UGV of mass to manipulate an object of mass , length , inertia , and whose center of mass is located at from the UGV. The saturations of the UAV torque and the UGV force are set to and , respectively. Additional viscous frictions are added to the model. The coefficients of viscosity for each general coordinate , , and are , , and , respectively. These parameters have been identified on the experimental testbed presented in the next section. The system is controlled by (4.1), (4.2.1), (4.2.2), and (4.2.2), where the control gains are , , , , , , and . Simulations are carried out considering the initial condition with the desired object pose set to and . Fig. 3 provides the numerical comparisons given:

  • No feedforward: The maximum thrust of the UAV is set to . The outer loop is controlled by (4.2.2) and the closed-loop is subject to a direct step variation of the desired reference.

  • Feedforward: The maximum thrust is set to . The outer loop uses the feedforward (16) instead of (4.2.2), and the closed-loop is subject to a direct step variation of the desired reference.

  • No RG: The maximum thrust is reduced to . The outer loop uses feedforward (16) and the closed-loop is subject to a direct step variation of the desired reference.

  • RG: The maximum thrust is set to and the outer loop uses the feedforward (16). The applied reference is issued by the RG using bisection with sampling time and a prediction time horizon .

Fig. 3 shows that the control law (4.2.2) stabilizes the system to the desired references but may trigger undesired behaviors such as high overshoots, which can be problematic in the case of less capable UAVs as clarified later on. We thus illustrate the advantage of using the feedforward action (16) instead of the proposed law (4.2.2). In particular, we see that using directly the relative angle instead of the desired angle to compute can improve the damping performance of the controlled system. Using the ISE/IAE performance indices [14] for the error on , we obtain without feedforward, and with feedforward.
For what regards constraints satisfaction, we compare the response with/without RG to illustrate the utility of implementing the RG in the case of less capable UAVs (i.e., for lower ). Without the use of the RG, we observe in Fig. 3 that the system violates the constraints i.e., the object angle goes beyond making the object fall down to , from which the system cannot recover. This is why, the RG is implemented and we observe from Fig. 3 that the system trajectories move safely to the desired reference without violating constraints.

8 Experimental Results

The experimental testbed (see Fig. 4) consists of a birotor UAV, a carbon rod, and a moving cart.

Fig. 4: Experimental setup.

The encoder Scancon SCA16 measures the relative attitude of the UAV , the Hengstler Incremental Push Pull Rotary Encoder RI58-0 measures the angle of the rod , and the Hohner encoder measures the object position . The Brushless Controller Simon Serie are electronic speed controllers that control the speed of the UAV propellers, which are brushless DC motors. The variable-frequency drive Junus JSP-090-20 is used to control the Parvex Axem motor of the worm drive system. The control algorithms are implemented through dSpace and Simulink. The system is controlled using (4.1), (4.2.1), (4.2.2), (4.2.2), and (16) using the same parameters as the simulations. Note that a saturated integral term with gain has been added to the outer loop to reject the steady-state disturbances induced by some of the neglected aspects, such as the additional gravity term induced by the cables at different positions.

Fig. 5: Comparison between numerical results and experimental results. The solid lines represent the states of the system and the dashed lines represent the applied reference.

The desired references for the object inclination and position are set to and , respectively. Fig. 5 shows that the proposed control law stabilizes the system to the desired references. Furthermore, we show that the data from the simulated model fits well the real data. The videos of the experiments can be found on https://wp.me/p9eDF3-3W.

9 Conclusions

This paper proposes a control scheme to position and orient an object by means of a UAV and a UGV. In particular, this paper proposes a control scheme where the UAV is tasked with the control of the object inclination, whereas the UGV is tasked with the control of the object position. Small gain arguments are used to prove asymptotic stability of the points of equilibrium. An RG unit is then added to the pre-stabilized system to augment the basin of attraction of the points of equilibrium. Numerical simulations and experimental results are provided to demonstrate the effectiveness of the proposed control scheme.

Acknowledgements

We would like to warmly thank Mr. Serge Torfs of our department for his help in the construction of the testbed.

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A Basic properties of system (2)

Lemma 7

is positive-definite and symmetric.

PROOF. It is immediate that is symmetric. is also positive definite since is Hermitian and all its eigenvalues are strictly positive.

Lemma 8

is skew-symmetric.

PROOF. The time derivative of is

(A.1)

Then, following from (A.1), we obtain

(A.2)

which is anti-symmetric and, therefore, skew-symmetric.

B Proof of Lemma 3

Using triangular inequality, satisfies

(B.1)

First note that is clearly bounded by

(B.2)

since for any , and for any . Furthermore, we can say that

(B.3)

since and for any , and for any .

For what concerns in (B.1), following from (4.2.2), we have

(B.4)

For , due to the saturation , we have that since .

As for restricted to , it is easy to see that (B.4) is continuous as the only potential singularity admits a finite limit, which is

(B.5)

Since is continuous and differentiable in the closed interval , the possible extrema of (B.4) can be found at the boundaries and at the stationary points, where . The only point where is . Therefore, since and , Eq. (B.4) reaches its maximum when (see Fig. B.1).

Fig. B.1: with and .

In particular, since is strictly decreasing for , where is defined as in (4.2.2) and is a function of , the maximum of is .

Consequently,

(B.6)

for any , and for any since for any . Furthermore, since , we can deduce that

(B.7)

for any , any , and any .

Following from (B.2), (B.3), (B.6) and (B.7), Lemma 3 holds true, concluding the proof.

C Proof of Lemma 4

The first part will define the total energy , which has useful properties to prove ISS and will be used to define the globally positive definite Lyapunov-candidate-function . The second step is to prove that is strictly negative with restriction and whenever the norm remains outside a ball of radius . The last part will be devoted to prove that there exists a finite asymptotic gain between and .

Define the total energy ,

(C.1)

Introducing (C.1) in (5), can be rewritten as

(C.2)

In the following, will be restricted to

(C.3)

where is any arbitrarily small positive value. Denote and as the smallest eigenvalues of and of , respectively. These eigenvalues are

(C.4)
(C.5)

The two following lemmas highlight two important properties of that will be used to prove ISS.

Lemma 9