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Research Article | Open Access
Bibhya Sharma, Jito Vanualailai, Avinesh Prasad, "A -Strategy: Facilitating Dual-Formation Control of a Virtually Connected Team", Journal of Advanced Transportation, vol. 2017, Article ID 9213805, 17 pages, 2017. https://doi.org/10.1155/2017/9213805
A -Strategy: Facilitating Dual-Formation Control of a Virtually Connected Team
This paper describes the design of new centralized acceleration-based controllers for the multitask problem of motion planning and control of a coordinated lead-carrier team fixed in a dual-formation within an obstacle-ridden environment. A -strategy, where and are Euclidean measures with respect to the lead robot, is developed to ensure virtual connectivity of the carrier robots to the lead robot. This connectivity, built into the system itself, inherently ensures globally rigid formation between each lead-carrier pair of the team. Moreover, a combination of target configuration, -strategy, orientation consensus, and avoidance of end-effector of robots results in a second, locally rigid formation (not infinitesimally rigid). Therefore, for the first time, a dual-formation control problem of a lead-carrier team of mobile manipulators is considered. This and other kinodynamic constraints have been treated simultaneously via the overarching Lyapunov-based control scheme, essentially a potential field method favored in the field of robotics. The formulation of this new scheme, demonstrated effectively via computer simulations, is timely, given that the current proposed engineering solutions, allowing autonomous vehicles on public roads, include the development of special lanes imbued with special sensors and wireless technologies.
1.1. Motivational Work
Researchers have continually designed and introduced numerous robotic systems that have a large array of possibilities for real world applications. One such system of increasing practical importance is based on swarm intelligence, which is a coordinated robot collective that is frequently sighted in airports, factories, wharfs, farms, and mines  and will play a huge role in the future as well. Swarms or flocks in nature are commonly dovetailed with formation types and unique patterns, seen, for example, in schools of fishes, flocks of birds, swarms of insects, and herds of animals. In addition, a wide spectrum of the “formation rigidity” has also appeared in literature using the nomenclature from [2, 3]; on one end there are split/rejoin maneuvers also known as minimally rigid formations [2, 3] which are required in applications such as reconnaissance, sampling, and surveillance, while on the other end there are globally rigid formations which are required in applications that require cooperative payload transportation [4–6]. Then there are locally rigid formations (required in convoying and demining [1, 7]) which are not infinitesimally rigid and allow for slight distortions temporarily. A dual-formation would be accomplishing two different formation types or patterns in parallel from a team in motion. However, it is noted that motion planning and control of a dual-formation are usually avoided due to the complexity of the algorithms and associated computer simulations and roll out to experimental designs.
Formation control can be defined as controlling the configuration or state of a team of agents en route to a target, normally maintaining constant their relative locations, hence maintaining the formation type and pattern [8–11]. Indeed, scalable formations with increased heterogeneity are evidently deployed to satisfy stringent time, labor, and cost demands. The literature contains rich and diverse research on motion planning and control of such collectives, which are ideally suited and capable of performing wide-ranging tasks such as those mentioned above. These can take place in known or partially known environments; an environment may be harsh or hazardous or even inaccessible to humans . Collectives provide increased performance, redundancy, flexibility, robustness, and multibehaviors which is difficult, and at times impossible, with single agents [12–14].
A number of approaches such as leader-follower, virtual structure, nearest neighbors, social potentials, behavior-based, and formation-constrained functions have been utilized to address formation control . While the lead-carrier approach is known for its poor disturbance rejection properties  and dependence placed upon a single agent, it is commonly utilized in literature because of its simplicity, scalability, and ability to contain an array of formations with richer specifications and complexities [3, 9, 11, 15–17]. This work deploys a new lead-carrier scheme to establish and control a team fixed in a dual-formation, for the first time.
1.2. Mechanical System
Of the many mechanical systems recorded in literature, mobile manipulators play a pivotal role in most industries that require a certain degree of automation and repetition. The literature takes into account the type of mobility of mobile manipulators with four possible configurations: (1) Type () where both platform and manipulator contain holonomic constraints, (2) Type where the platform is holonomic and manipulator is nonholonomic, (3) Type where the platform is nonholonomic but the manipulator is holonomic, and (4) Type where both the platform and the manipulator contain nonholonomic constraints.
In this paper we consider Type which was introduced by Seraji in  in 1998 and classified as the nonholonomic mobile manipulators. Since then the literature has grown with research using Type mobile manipulators (see, e.g., [11, 19–21]). The challenges associated with motion planning and control of the Type are amplified by the intimate coupling of the nonholonomic and holonomic constraints arising from the amalgamation of an articulated robotic arm and a wheeled platform.
1.3. Lyapunov-Based Control Scheme
The Lyapunov-based control scheme (LBCS), proposed by Sharma et al. in [3, 11], can be categorized under the artificial potential field method commonly deployed in the area of robotics research, especially applied to motion planning and control of various robotic systems [1, 3, 11]. We utilize the control scheme to derive centralized acceleration-based controllers for a dual-formation control of a team of lead-carrier 2-link mobile manipulators.
The seminal idea behind LbCS is to design an appropriate Lyapunov function which acts as total potentials. LbCS entails the construction of attractive and obstacle avoidance functions for the attraction to target and repulsion from various obstacles, respectively. While the attractive functions can be treated as attractive potential field functions, per se, the repulsive potential field functions are designed as ratios with the obstacle avoidance function in the denominator and a positive tuning parameter in the numerator. The sum of these potential functions is termed as the total potentials, a basis to design the controllers for the team. The governing principle behind the method is to attach attractive field to the target and repulsive field to each obstacle. The whole workspace is then inundated with positive and negative fields, with the direction of motion facilitated via the notion of steepest descent. For the vehicular systems, the gradient of the total potentials, referred to as the input force, determines the speed and the direction along which the vehicle moves.
The Lyapunov-based control scheme utilized in this paper has been introduced in numerous applications for various robotic systems including ones tagged with holonomic or nonholonomic constraints [3, 22]. It is relatively easier to construct mathematical functions from limitations, inequalities, restrictions, and mechanical constraints tagged to the robotic systems and incorporate them in the controllers derived from LbCS, compared to the other motion and control schemes from literature [3, 23, 24]. For example, amongst others it is very difficult to capture the dynamical constraints (such as limitations on steering angle) of a system into the controllers, but LBCS has an in-built process of converting these into artificial obstacles and incorporating them into the nonlinear controllers.
While many papers have considered motion planning and control of Type formation, this paper adds a new concept called virtual connectivity to formation control. This is where all the carrier robots are virtually connected to the lead robot along the trajectory to task completion. The concept inspired by the work by Consolini et al. in 2008 in  and Sharma et al. in 2012 in , is classified as the -strategy where and are Euclidean measures of a lead-carrier robot scheme. Fixing the values of enables us to maintain a fixed distance between the lead robot and each of its carriers. In addition, enables orientation change of the carriers which will optimize avoidance and the lane change and lane merge maneuvers.
In this research, the team of robots is virtually connected, meaning that all agents are bounded mathematically as an automated and intelligent swarm in formation. While connected vehicles can normally be seen to facilitate the Vehicle-to-Vehicle (V2V) and the Vehicle-to-Infrastructure (V2I) communications, collectively defined as Vehicle-to-Everything (V2X) communications, on our roads more recently through wireless technologies such as cellular networks, this research takes a step towards possible applications of the connected vehicles through the dual-formation of an autonomous swarm in a obstacle-ridden environment, such as heavy-traffic highways with multiple lanes. In order to recognize this difference, we consider the swarm to be virtually connected and behave as an autonomous and intelligent robot system, controlled via the Lyapunov-based control scheme. Another advantage of the virtual connectivity is the fact that the lead-carrier team have the capability to satisfactorily complete diverse tasks which is not possible with merely cooperative agents. Typical examples of the connected mobile manipulators are the freight vehicles which are responsible for payload transfer, transportation of large objects, and transportation of multiple objects simultaneously on highways and roads, with safe and collision-free motion through vehicular internetworking and V2I communications. While the swarm intelligence ensures energy efficiency and reduced costs, the new dual-formation ensures multitasking, job precision and another solution to applications on roads such as convoying and payload transfer.
It is the authors’ belief that the problem of maintaining a dual-formation of a lead-carrier team with virtual connectively within an obstacle cluttered environment such as a heavy-traffic highway is treated for the first time within a framework of LbCS. While global rigidity of formations is not possible with nonholonomic constraints, we maintain local rigidity of the team in formation with reference to the end-effectors and also establish a globally rigid formation of every lead-carrier pair in the team. The overarching framework is a lead-carrier scheme to establish, maintain, and translate the whole team under dual-formation through centralized control laws.
Finally, treatment of various categories of obstacles is included within the research problem. For connected and automated vehicle system on roads and highways, stationary obstacles can be treated as road pavements or lane boundaries which need to be avoided in order to contain and maintain individual or multiple vehicles in designated road lane(s). As an application, we consider a number of rigid-shaped objects needed to be dropped off at precise coordinates and bearings, en route a collision and obstacle-free trajectory. This is very useful in situations involving loading/offloading on docks, mines, and military bases and the lane changing and maneuverability on roads and highways, where precision is paramount.
2. System Modelling
Let be the th 2-link mobile manipulator (2MM) consisting of a car-like wheeled platform (front-wheel steered) with a 2-link planar arm mounted on the midfront axle of the wheeled platform. and for are the lead and carrier robots, respectively, of a team of 2MMs.
The th articulated body of the 2MM, , is a disk with radius and is positioned at center and it represents the set . Note that represents the wheeled platform while represent links 1 and 2, respectively (see Figure 1). Precisely, the th 2MM is the set given by .
The dimensions of each 2MMs are kept the same; that is, for . The maximum speed and maximum steering angle of each mobile car-like platform are also kept the same.
The planar workspace is a fixed, closed, and bounded rectangular region defined for some and , as
With reference to Figure 1, gives the location of the center of the wheeled platform of , while is the steering angle with respect to the platform’s longitudinal axis. Now, gives the platform’s orientation with respect to the -axis, gives the orientation of Link 1 with respect to its platform, and gives the orientation of Link 2 with respect to Link 1. For simplicity, we let .
Note the presence of clearance parameters for safety of the wheeled platform and the gripper . The position of the end-effector of with respect to its wheeled platform can be written as One can comfortably show that the dynamic model of for with respect to the end-effector is
The reader is referred to  for a detailed derivation of the system of ODEs. We see that and are the instantaneous translational and rotational accelerations, respectively, of the wheeled platform, while and are the instantaneous angular accelerations, respectively, of the lower and upper links of , the latter being relative accelerations since the torque and moment of inertia of the system are considered as constants. In addition, we assume no slippage (i.e., ) and pure rolling (i.e., ) of the platform wheels. These nonintegrable constraints, denoted as the nonholonomic constraints, are captured in system (2).
To ensure that the complete safely steers past an obstacle, we adopt the nomenclature of  to enclose each body of by the smallest possible circle. Given the clearance parameters (see Figure 1), we enclose the wheeled platform by a protective circular region with radius , Link 1 with radius , and Link 2 with radius . This procedure provides the 2MMs with maximized free space.
Furthermore, the positions of the articulated bodies of the th 2MM can be expressed completely in terms of the state variables , , , , and . Hence, for the articulated bodies of we ascertain where is a floor function. These position constraints are known as the holonomic constraints of the 2MM system .
Note that for the target attraction and obstacle avoidance functions that will be defined in later sections, we consider for and for the th articulated body of .
3. Main Objective
The main objective is to design artificial potential functions (APFs) from LbCS described in [1, 3, 25, 26] and, accordingly, derive centralized acceleration-based controls such that the lead-carrier pairs in the team, represented by system (2), can transfer rigid rod-shaped loads, which approximate the desired orientations of the loads and the carriers. In addition, to ensure that the carriers move cohesively in a prescribed pattern and do not collide with each other, their end-effectors will also be fixed in a locally rigid formation. We work with a random number of carriers displaced from the lead robot. That is, we let set as the set of distances between the lead robot and the carriers.
While each lead-carrier robot pair is maintained in a globally rigid formation (global rigidity) to carry rigid objects precisely, we also want the the whole team to be fixed in parallel within a locally rigid formation (local rigidity) to ensure an overall pattern for cohesiveness, safety, and specific task-requirements.
The design of the nonlinear control laws is captured in Figure 2, clearly illustrating the roles of the new control scheme and the -strategy.
4. Globally Rigid Formation of Lead-Carrier Pairs
A globally rigid formation is a configuration in which distance and angle between the end-effectors of the lead-carrier pairs are strictly maintained en route to the destination. To ensure globally rigid formation of every lead-carrier pair and their cohesiveness, we propose a new strategy which we now classify as the -strategy.
The -strategy is inspired by the work carried out by Sharma et al. in 2014 in  and Consolini et al. in 2008 in . In the latter, hierarchical formations were achieved for unicycles with input constraints on the admissible trajectories of the lead robot guiding the formation. The shape of the formation also changed according to the motion of the lead robot. The advantages of the -strategy over earlier leader-follower schemes from the authors [3, 26] are the following: (1) it contributes to the cohesion of the team; (2) the lead robot needs not be positioned in front of the pack; (3) there is a need for velocity consensus of the team; (4) there is no need to design separate attractive functions for the carriers, with the function established for the lead sufficing; and (5) rotation(s) of the formation is enabled, if needed, during avoidance maneuvers.
The -strategy establishes a virtual connection between the lead and each carrier of the team. This ensures that the lead-carrier distances required for the rigid formation are fixed and maintained at all times. Specifically, the strategy requires that the end-effector of the th carrier robot be fixed via parameters and with respect to the end-effector of the unconstrained leader (see Figure 3). While is the length of the line from the end-effector of the leader to the end-effector of the th carrier robot of the team, is the angle to the line with respect to the main frame for .
Now, with reference to Figure 3, we have which is true for , considering all possible orientations in the workspace. The addition formula for cosine gives which differentiated as Then the ODEs governing the dynamic model of 2MMs, incorporating the -strategy, and hence ensuring virtual connectivity between all lead-carrier pairs can be rewritten as where for . We note that corresponds to the lead robot and if system (7) basically collapses into system (2), which governs the motion of 2MMs not constrained by any connectivity. System (7) is a description of the instantaneous velocities and accelerations of the various bodies of . We assume that the instantaneous accelerations , and can move the end-effector of , within the framework of virtual connectivity, to the final designation. Ultimately for are considered by the LbCS as the nonlinear acceleration controls of the 2MMs. Let the vector refer to the position of the end-effector of , the orientations of the various components of , and the velocities of the various components of at time . Let and . Then system (7) can be written compactly as where is the matrix Let refer to the positions, orientations, and the velocities for MMs, . Let and . Then we have the following initial-value problem for MMs: where if is the matrix of all zero entries,Now, assume that the final position of the end-effector of is at the point and final orientation at this point is . Its final instantaneous velocity vector is . Then the points, are the components of the equilibrium point of system (12), with
5. Locally Rigid Formation of the Team
A locally rigid formation is a configuration in which all the interrobot distances in a prescribed formation are maintained en route to the destination; however, temporary slight distortions are allowed. That is, the formation gives the end-effectors freedom to distort from their arrangement in the overall team pattern temporarily to be able to carry out a task effectively, such as collision avoidance of obstacles.
Our work requires a coordinated transfer of loads by the lead-carrier team. Therefore, we need the team to be fixed in a locally rigid formation at all times . The purpose of this is threefold: (1) it prevents the end-effectors from colliding or drifting off, therefore ensuring cohesion, (2) it maintains a prescribed formation which heralds order and purpose, and (3) it ensures that all lead-carrier pairs are contained in a neighborhood. We note that only the lead robot has the global information of the workspace, which includes the target configuration; thus, the carrier robots are guided by the lead robot. This implies the centralized nature of the problem.
Inspired by the work of Li and Xiao , Schneider and Wildermuth , and Sharma et al. , we introduce a new design in order to establish, maintain, and translate a locally rigid formation during the motion. This design is an amalgamation of the following modules: (1) interrobot bounds, (2) target configuration, and (3) orientation consensus, which will be considered separately in the next sections. We will design the repulsive and attraction functions (which will be part of a Lyapunov function) to elucidate the importance and contribution of these modules to locally rigid formations.
5.1. Interrobot Bounds
5.1.1. Minimum Interrobot Bound
To prevent any possible collisions between the end-effectors we shall adopt the following obstacle avoidance function from : where is the minimum Euclidean distance between end-effectors of and on .
5.1.2. Maximum Interrobot Bound
The relative distance between the end-effectors of any two robots needs to be bounded. Accordingly, we design an obstacle avoidance function where is the maximum Euclidean distance between end-effectors of and on . These bounds are treated as artificial obstacles. To avoid these obstacles we design new repulsive potential field functions. The reader is referred to  for a detailed account of the relationship between bounds, artificial obstacles, and avoidance functions.
To generate repulsive effects from (16) and (17) we design the corresponding repulsive potential field function which in essence is a ratio that encodes the avoidance function in the denominator and a positive tuning parameter in the numerator. Manipulation of the tuning parameters associated with functions and provide an added degree of control and help maintain the locally rigid formation.
Figure 4 illustrates the total potentials generated from the attractive potentials produced by (19), and the repulsive potentials produced from and to observe the maximum and minimum bounds between robots and .
Henceforth, for each obstacle (fixed, moving, or artificial), we will construct an appropriate avoidance function and, in accordance with the LBCS, design similar repulsive potential field functions to generate the collision and obstacle avoidance maneuvers. All-in-all the avoidance capability of the control scheme lies in the creation of repulsive potential functions that induce an increase or decrease in the instantaneous rate of change of total potentials.
5.2. Target Configuration
This work aims to imitate a real-life application wherein a number of rigid-shaped objects need to be dropped off simultaneously at precise coordinates and bearings, which is very useful in applications such as loading/offloading on docks or parking bays where precision is paramount. This translates to the control of the final configurations of the end-effectors.
First, we need to affix a target for , with respect to its end-effector, to reach after some time . A possible target with center and radius can be
Second, the load is to be presented (deposited or dropped off) at the target with a precise orientation. Since we have the ends of the load held and carried by the end-effectors, there is a need to consider the final positional coordinates and orientation of each end-effector. However, with the introduction of the -strategy there is no need to establish separate attractive functions for the positional coordinates of the carriers, illustrating the centralized nature of the controllers.
To establish an attraction to the final configuration of and the overall final bearing of the team, we consider attractive potential field functionsfor the lead and the th carrier, respectively. The functions are positive for all . Note that are the final orientations of the wheeled platform, Link 1 and Link 2, while are the angle-gain parameters, which have a value of if a final orientation is warranted, or as the default value . Then in the LbCS, (19) will act as attractors by having the team move to the target configuration and ensure that system trajectories start and remain close to a stable equilibrium point of system (7).
5.2.2. Mobile Platforms
In situations that reflect the objectives of this work, we observe that the mobile platforms of the carriers are parked at specific coordinates and bearings such as the parking bays. Since the end-effectors converge to the final target with the desired orientations of the links, the positional requirement of the mobile platforms is already captured in (19). Therefore, we need to consider only their final orientations and thus complete fixing the overall bearing of the team. For this we adopt the minimum distance technique (MDT) and the concept of ghost parking bays from [3, 26]. Ghost parking bays are constructed with precise coordinates in the neighborhood of the final configuration of each mobile platform, like caging with a single entrance.
Utilizing MDT, we identify the closest point on each boundary line (treated as a line segment) of a ghost parking bay measured from the CoM of the wheeled platform of . The underlying assumption of the technique is that avoidance of the closest point ensures avoidance of the entire line segment at every iteration . Now, the parametric representation of this th line segment with initial coordinate and final coordinate where is the parameter is Minimizing the Euclidian distance from to the th line segment, we get where
while we utilize a saturation function given as We note that a specific ghost parking bay needs to be fixed for each 2MM; hence, a mobile platform will be surrounded by three boundary lines which have to be avoided. This means that the th mobile platform will be avoiding , , and boundary lines. For avoidance, we consider a measurement of the distance between and the closest point on th boundary line To generate repulsive field around these ghost parking bays we again design new repulsive potential field functions, as discussed in the previous section.
5.3. Orientation Consensus
An important feature of locally rigid formation navigation of the robot team is its orientation consensus or, perhaps more frequently, being cited in literature as common heading. Potential functions which facilitate an orientation consensus are needed to be incorporated into the total potentials to provide the connected swarm or team with a common heading and to ensure that the mobile platforms are all orientated in the same direction. A failure of the orientation consensus not only unnecessarily upsets the locally rigid formation but also compromises the overall purpose of the transportation of transferring motion of the team. For example, the payload transportation or multiple-load transfer can be difficult or impossible if the mobile bases are not aligned and travelling on different headings with respect to the lead robot.
In this work, the orientation consensus with reference to the mobile platform is established by the attractive function: This function is also responsive to the cohesion of the team where the lead’s orientation matches the average orientation of the team. The function stands well with the overall centralized architecture of the controls in this work, although the literature is rich with forms that contribute to distributed controls.
6. Integrated Subtasks
We further include in this work a number of kinodynamic constraints from the workspace or ones tagged to the 2MMs.
6.1. Avoidance of Stationary Obstacles
Let us fix stationary obstacles within the boundaries of the workspace. We assume that the th stationary obstacle, for , is circular with center given as and radius and defined as . Note that it is only for simplicity of analysis and illustration of the Lyapunov-based methodology that we have chosen circular obstacles. However, any convex polygonal obstacle can be considered in our methodology because we can apply the MDT (minimum distance technique) that we utilized in Section 5.2.2 to ensure avoidance between the end-effector and the nearest point of a line segment.
For the avoidance of the stationary circular obstacles we construct separate avoidance functions for each body of : For the vehicle system on roads and highways, these stationary obstacles can be treated as road pavements or lane boundaries which need to be avoided in order to contain and maintain individual or multiple vehicles in designated road lane(s). The road infrastructure is assumed to be well equipped with sensors and wireless technologies to ensure adequate and timely Vehicle-to-Infrastructure (V2I) communication for the real-life applications of the automated and intelligent vehicle system to work. The assumption is applicable keeping in mind that the current proposed engineering solutions (to the problem of allowing autonomous vehicles on public roads) include the development of special lanes on roads purely for self-driving vehicles imbued with the special sensors and wireless technologies.
6.2. Avoidance of Moving Obstacles
Each solid body of an articulated 2MM has to be treated as a moving obstacle for all the other 2MMs in WS. Avoidance of the end-effectors is already captured in the recipe to generate and maintain the locally rigid formation. Therefore, for the platform of to avoid the platform of , we shall use an avoidance function: Mechanical singularities and bounds on velocities are treated as dynamic constraints. In practice, bending angles of the links are limited due to the mechanical singularities, while the velocities of the links and the wheeled platform are restricted due to safety reasons . In accordance with the LbCS, each dynamic constraint is treated as an artificial obstacle and appropriate obstacle avoidance function designed for the avoidance.
6.3. Mechanical Singularities
Singular configurations arise when we have the following.(i), , or . Subsequently, the condition placed on is for , which implies that Link 2 can neither be fully stretched nor be folded back.(ii)The angle between Link 1 and the platform is bounded by . Simply worded, Link 1 of the th 2MM can only freely rotate within .(iii)Due to the inclusion of the -strategy, singularities arise when . To avoid this we should include , for .For avoidance of these singularities, the following obstacle avoidance functions will be included: These positive functions would guarantee a strict observation of the mechanical singularities of the 2MMs when encoded into specific repulsive potential field function.
6.4. Modulus Bound on Velocities
From a practical viewpoint, the translational and rotational velocities of the 2MMs are limited, so we include the following constraints:(i), where is the maximal achievable speed.(ii), where . This condition arises from the boundedness of the steering angle, . That is , where is maximal steering angle.(iii) and , where , are the maximal rotational velocities of Link 1 and Link 2, respectively.We construct avoidance functions so that can successfully avoid the artificial obstacles created from the constraints above: for . These positive functions guarantee the adherence to limitations when encoded appropriately into the repulsive potential field functions.
7. Controller Design
We now define a Lyapunov function and subsequently extract from it the nonlinear control laws for system (7). However, to guarantee that the total potentials vanish precisely at the equilibrium state, we design an auxiliary function that would be multiplied to the repulsive potential field functions.
7.1. Auxiliary Function
7.2. Lyapunov Function
Combining the attractive and repulsive potential field functions and introducing tuning parameters, and for , we consider a Lyapunov function for system (12) (suppressing ):
7.3. Nonlinear Acceleration Controllers
We utilize LbCS  to derive the acceleration-based controllers. It is easy to see that the Lyapunov function in (31) is continuous and positive and has continuous first partial derivatives on the domain: Moreover, vanishes at the equilibrium point . Let , , , and . The time derivative of , along a solution of system (12), is provided we set the control laws aswhere , , , for and are constants classified as convergence parameters and the functions and are defined as
7.4. Proof of Stability of Equilibrium Point
We note that LbCS produces feedback controllers, which depend explicitly on the state variables, , and hence implicitly on time, . That is, , , , and . Thus, we can let .
Proof. The higher-order partial derivatives of , , , and are continuous on for the simple fact that functions that appear in the denominator of these functions are in and will also appear in the higher-order partial derivatives of the controllers. Since the other ODEs in (12) also have higher-order partial derivatives in , with each continuous over , we have that ; that is, is locally Lipschitz on . This implies there are unique solutions of system (12) in on some finite time interval . To prove their stability, we see that