| The unilateral constraint denotes a mechanical constraint which prevents penetration between two bodies.In this paper,we focus on the unilateral constraint caused by a rope.The multi-body aircraft system with unilateral constraint is an aircraft connected with other rigid bodies via flexible rope.This kind of system is widely used in both military and commerce.Thereinto,UAV with slung-load can apply to rapid transportation,proximity detection,fire suppression and some military missions,such as demining and surveillance.Multi-body flight system with a payload connected by flexible cable retains the maneuverability of the carrier UAV,and relaxes the restriction of payload size.Hence the multi-body system can fulfill the task in a complex environment,which improves the applicability of the system.However,the rapid motion of carrier UAV causes a swing of slung-load seriously,which will degrade the control performance of the system.Sometimes,it makes the system unstable.Then solving the swing-free tracking control problem of the multi-body aircraft system with unilateral constraint becomes the primary issue of the air transportion with slung-load UAV.This work takes a slung-load quadrotor system as research object,and then the slung payload swing-free tracking problem is studied in differential geometry viewpoint.According to this,it mainly includes following respects:geometric modeling of the multi-body system,intrinsic geometric control method,system configuration manifold observation,estimation of the velocity vector in the tangent space,and trajectory generation in the tangent bundle.First of all,a hybrid geometric model of the multi-body system is provided in a geometric viewpoint,and local controllability along a feasible trajectory of the system is investigated.It provides a basis for follow-up research.A hybrid dynamic system is utilized to describe the multi-body system with unilateral constraint.Then the multi-body system is decomposed to a dynamic system with zero cable tension and a dynamic system with nonzero cable tension.A series of discrete switching events are used to describe the interaction of the states of two sub-systems.The Lie group and homogeneous manifold are used to describe the configuration manifold of subsystems,and then Hamilton least-action principle is employed to obtain the geometric dynamic model.On this foundation,more compact formulations are provided via intrinsic equations with Levi-Civita connection.Besides,geometric local controllability is studied via infinitesimal variations along a feasible trajectory.Then the rotor-propeller system with the constant voltage,the relation between rotation speed and thrust are investigated.Second,robust sliding-mode control structures on SO(3)andS~2 are provided to build a swing-free tracking controller for the slung-load.This work is necessary to realize swing-free trajectory tracking.With the singular perturbation theory and cascade control method,a tracking controller structure for the dynamic system with nonzero cable tension is presented.Then the tracking control problem on a high dimension manifold is translated into an integral curve tracking problem on decreased dimension sub-manifold.Considering the convergence time of attitude configuration SO(3)and uncertainties,a nonsingular terminal sliding surface and super-twisting method are used to design an almost-global finite-time controller on SO(3).Then existence and impaction of critical points on SO(3)are discussed and a hybrid configuration error function is built to produce the hybrid error on SO(3).Finally,a global finite time convergence controller on SO(3)is obtained.To solve the payload azimuth configuration tracking problem onS~2,a sliding mode controller is developed onS~2,which extends the conventional sliding mode method and reaching law from flat Euclidean space to the non-flat manifoldS~2.Besides,a payload position robust tracking controller is provided,and then tracking controller synthesis is achieved.Under the condition that carrier UAV attitude configuration is near equilibrium states,pay-load azimuth configuration and position are convergent asymptotically,which is strictly and theoretically proved.Afterward,motion acceleration compensation method and an adaptive explicit complementary filter are provided to improve the precision of attitude configuration reconstruction considering motion acceleration.It provides necessary intrinsic information to the geometric feedback controller.In this part,the carrier UAV’s attitude configuration manifold construction problem is studied.An attitude configuration manifold constructed with a single sensor contains noise and bias of the sensor,which may lead to the great difference between real attitude configuration and constructed configuration.To solve this problem,an explicit complementary filter on SO(3)is used to reconstruct the attitude configuration of the carrier.Then,the effects of motion acceleration of carrier UAV on results of attitude configuration reconstruction are also discussed,and motion acceleration compensation framework is presented.The compensation framework uses the motion acceleration provided by the extra sensors to modify the gravity field,then the modified vector field is used in ECF.The proposed method decreases the bad influence caused by motion acceleration.Considering the compensation error and influence of magnetic meter caused by motor speed changing,adaptive functions are provided to adjust the weights of sensor errors online,and AECF is provided to improve the accuracy and reliability of the attitude configuration reconstruction method.Then,a robust intrinsic observer on high-dimension manifold is built to estimate the configuration and corresponding speed with intrinsic information of configuration manifold.This work provides necessary states for the system to full states feedback geometric controller,which can improve the reliability and applicability.To solve the problem that the configuration manifolds of the multi-body system are obtained without the corresponding speed on tangent space,a super-twisting intrinsic observer for a class of mechanical systems on Riemann is provided with the Riemann connection,metric tensor,parallel transportation,and curvature tensor.The intrinsic observer uses the intrinsic information of the system,which not depends on local coordinates.Lie group SO(3),homogeneous manifoldS~2,and Euclidean space~3all have special geometric structures.With their special geometric properties,super-twisting intrinsic observers on SO(3),S~2 and~3are provided respectively.Then the proposed super-twisting intrinsic observers are applied to build a multi-time-scale intrinsic observer for the multi-body system.the observer can estimate the carrier rotation speed,payload azimuth speed,and payload motion velocity rapidly and precisely.Last,differential-flatness properities of the multi-body system and trajectory generation method for under-actuated system are provided.With the proposed method,a feasible trajectory on tangent bundle is generated,which is the essential guarantee of slung-load swing-free tracking.To generate a feasible trajectory of multi-body in high-dimension tangent bundle,the differential-flatness properties of the multi-body system are obtained,and then the relation between the states on the tangent bundle and the flat outputs are provided.With the differential flatness method,the obstacles,state constraints,and control constraints on configuration space are mapped to the flat output space.Then the trajectory generation problem on high-dimension tangent bundle is converted to a trajectory generation problem on low dimensional flat output space.Then piecewise polynomial is employed to parameterize the trajectory on flat output space,and the trajectory generation problem becomes a nonlinear programming problem of the control points.The simulation indicates that the proposed method can generate a feasible trajectory for the multi-body system in the environment with known obstacles. |
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