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Continuous collision detection between A(t) and B. A(t) is a spline motion. p is a point on A. p's motion function is p(t) and v(t) = p (t) is the velocity. d is the closest distance between two objects and n is the direction of closest distance. τ = ∆t1 + ∆t2 is the first time of contact. ∆t1 and ∆t2 correspond to the advancements during two successive steps.
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We present a novel trajectory computation algorithm to smooth piecewise linear collision-free trajectories computed by sample-based motion planners. Our approach uses cubic B-splines to generate trajectories that are C2 almost everywhere, except on a few isolated points. The algorithm performs local spline refinement to compute smooth, collision-fr...
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... Subsequently, Ran et al. [19] further expanded this parabolic smoothing algorithm to incorporate jerk constraints. Furthermore, Pan et al. [20] introduced spline-based trajectory representation and its associated shortcut algorithm. Although all of these methods successfully compute smooth paths for piecewise linear trajectories, their computation speed is slow due to numerous collision checks. ...
In dynamic environments, real-time trajectory planners are required to generate smooth trajectories. However, trajectory planners based on real-time sampling often produce jerky trajectories that necessitate post-processing steps for smoothing. Existing local smoothing methods may result in trajectories that collide with obstacles due to the lack of a direct connection between the smoothing process and trajectory optimization. To address this limitation, this paper proposes a novel trajectory-smoothing method that considers obstacle constraints in real time. By introducing virtual attractive forces from original trajectory points and virtual repulsive forces from obstacles, the resultant force guides the generation of smooth trajectories. This approach enables parallel execution with the trajectory-planning process and requires low computational overhead. Experimental validation in different scenarios demonstrates that the proposed method not only achieves real-time trajectory smoothing but also effectively avoids obstacles.
... Sampling-based approaches are good candidates for high-dimensional planning problems, as they avoid explicit representation of the prohibited regions and instead incrementally sample the spacecraft's free state space, which significantly reduces the computational complexity [16,17]. But, the major deficiency in sampling-based approaches is providing non-optimal solutions since the sampling process in such methods terminates at a finite number of samples [18,19]. Therefore, a coupled and optimal approach it is worth investigating to handle the inspection motion planning problem. ...
Close proximity on-orbit inspection is a critical ability to initiate on-orbit servicing operations, required technology for space exploration missions. It is challenging to solve an optimal inspection trajectory planning problem that provides a complete target observation due to major difficulties. First, the inspection requirements must be defined and imposed on the problem as a set of path constraints that result in an NP-hard problem. Secondly, the optimization problem, including highly non-convex constraints, is very difficult to solve directly using an optimal control solver. Additionally, it requires proper initialization of states and control variables, which is critical in such problems. To overcome these difficulties, this paper proposes a novel formulation and method of solution for a full 6-DOF optimal inspection motion planning problem. Optimal inspection trajectories are designed for a rigid-body spacecraft, which performs close, continuous, and complete observation of rigid, non-rotating, and non-accelerating known targets. The inspection and collision avoidance constraints are defined in explicit forms that rectify the non-differentiability of the problem and satisfy the inspection requirements. A pseudospectral optimal control solver is implemented to numerically solve the trajectory optimization problem. The proposed methodology is applicable to any robotic inspection mission. Simulations are presented as a validation of the proposed methodology and the achieved optimality.
... This algorithm systematically identifies potential collision points along curves connecting these adjacent fitting points. Our collision detection process is guided by the principles articulated in references [39]. Should collisions be identified within the interval delimited by the two fitting points [t j , t j+1 ], we strategically introduce a new fitting point precisely at the midpoint of the original path spanning from t j to t j+1 . ...
Efficient logistics and transport at the port heavily relies on efficient AGV scheduling and planning for container transshipment. This paper presents a comprehensive approach to address the challenges in AGV path planning and coordination within the domain of intelligent transportation systems. We propose an enhanced graph search method for constructing the global path of a single AGV that mitigates the issues associated with paths closely aligned with obstacle corner points. Moreover, a centralized global planning module is developed to facilitate planning and scheduling. Each individual AGV establishes real-time communication with the upper layers to accurately determine its position at complex intersections. By computing its priority sequence within a coordination circle, the AGV effectively treats the high-priority trajectories of other vehicles as dynamic obstacles for its local trajectory planning. The feasibility of trajectory information is ensured by solving the online real-time Optimal Control Problem (OCP). In the trajectory planning process for a single AGV, we incorporate a linear programming-based obstacle avoidance strategy. This strategy transforms the obstacle avoidance optimization problem into trajectory planning constraints using Karush-Kuhn-Tucker (KKT) conditions. Consequently, seamless and secure AGV movement within the port environment is guaranteed. The global planning module encompasses a global regulatory mechanism that provides each AGV with an initial feasible path. This approach not only facilitates complexity decomposition for large-scale problems, but also maintains path feasibility through continuous real-time communication with the upper layers during AGV travel. A key advantage of our progressive solution lies in its flexibility and scalability. This approach readily accommodates extensions based on the original problem and allows adjustments in the overall problem size in response to varying port cargo throughput, all without requiring a complete system overhaul.
... After a robot plans its initial path, it needs to smooth the generated path (Dragan et al. 2013). Pan et al. (2012) smoothed the path generated based on sample planning through spline interpolation, randomly selecting a series of points along the original piecewise linear path and constructing cubic B-splines to interpolate or approximate these points. Ratliff et al. (2009) proposed a continuous path refinement method based on covariant gradient technology, which used covariant gradient technology to improve the quality of sampled trajectories. ...
Path planning is an NP-hard problem in road network environments. Considering that the existing path planning algorithms mainly have the problems of low smoothness and low search efficiency in generating paths in large-scale complex environments, an improved rapidly exploring random tree (RRT) algorithm is proposed in this paper. First, the grid method is applied to model the road network environment, and the RRT algorithm based on adjacency expansion is proposed to search the initial path. Then, the strategies of identifying paths and eliminating redundant paths are adopted, respectively, to further optimize the selected paths. Experimental results show that, compared with other path planning algorithms, our algorithm can achieve faster convergence speed, shorter search path, and better smoothness in a complex map of the environment.
... They also often use B'ezier Curves and Bernstein polynomials, which provide continuous curvature in the presence of seven control waypoints. Efficient methods for obtaining smooth curves are polynomial splines [12], clothoids [13], generalized Cornu spirals [14],10,15] are the most widely used at present. Their factors are calculated from the coordinates of the five control waypoints, which ensure that the curvature of the path changes smoothly. ...
... Therefore, it is not always possible to implement them in real time on an onboard computer. In particular, planning the motion of a robot in a dynamic environment, and avoiding collisions with several dynamic obstacles, is still an unsolved problem due to the lack of time to generate a safe trajectory [10,11]. ...
... χ 1 is treated as a bounded disturbance (15). To design system (4), we used linear local stabilizing feedback to the design system (10). In order to satisfy the given constraints (18) for the design of system (17), we introduce the sigmoid bounded stabilizing local feedbacks and appropriate residuals ...
For an unmanned aerial vehicle (UAV) of an aircraft type, the problems of planning achievable trajectories as well as robust control under wind disturbances are considered. A computationally simple method for compiling a primary non-smooth 4D trajectory is proposed. Its segments connect the given waypoints and determine the desired average velocity in various sections. Instead of time-consuming methods of analytical smoothing of broken path joints using polynomials, a tracking differentiator with S-shaped smooth and limited sigmoid corrective actions is developed. This virtual dynamic model provides natural smoothing of the primary trajectory considering the design constraints on the velocity, acceleration, and thrust of the UAV. The tracking differentiator variables create an achievable trajectory and are used to synthesize the UAV tracking system. To compensate for the action of wind disturbances on the UAV, a disturbance observer is developed. It is a replica of the equations of the control plant model, which are directly affected by external uncontrolled disturbances. These algorithms also use sigmoid corrections. Unlike standard disturbances observers, this approach does not require the development of a dynamic model of external disturbances and does not assume their smoothness. The effectiveness of the developed algorithms was confirmed by numerical simulation.
... The problem of admissible trajectory planning is a separate, rather time-consuming task. As a rule, it is solved off-line, using spline interpolation or complex geometric calculations [17][18][19][20][21][22][23][24][25]. If there is reason to believe that the given curves are smooth and realizable, then one can solve the problem of restoring their derivatives and filtering by the above methods (using dynamic differentiators with special tuning). ...
... a priori. Its minimum value is determined by constraints (18) and the response of controlled variables during the processing of jumps. As a result, the output of the generator ) ( 1 t x will produce a path that is a smoothed analog of the useful vector signal ) ...
On the example of a control system for an unmanned aerial vehicle, we consider the problems of filtering, smoothing and restoring derivatives of reference action signals. These signals determine the desired spatial path of the plant at the first approximation. As a rule, researchers have considered these problems separately and have used different methods to solve each of them. The paper aims to develop a unified approach that provides a comprehensive solution to mentioned problems. We propose a dynamic admissible path generator. It is constructed as a copy of the canonical control plant model with smooth and bounded sigmoid corrective actions. For the deterministic case, a synthesis procedure has been developed, which ensures that the output variables of the generator track a non-smooth reference signal. Moreover, it considers the constraints on the velocity and acceleration of the plant. As a result, the generator variables produce a naturally smoothed spatial curve and its derivatives, which are realizable reference actions for the plant. The construction of the generator does not require exact knowledge of the plant parameters. Its dynamic order is less than that of the standard differentiators. We confirm the effectiveness of the approach by the results of numerical simulation.
... For general cases, collisions can be checked between the swept volume of an object with a specific custom bounding volume hierarchy [15]. Conservative advancement [16] is another approach, though it often gives too conservative results when objects are close together but not touching. For our method, it provides another analytical solution to CCD by transforming the collision conditions into a set of polynomial inequalities, whose roots can be efficiently solved by the proposed rootsfinding algorithm. ...
... In this case, we set the same end condition for quadrotor s g = [600, 650, 700, 0, 0, 0, 0, 0, 0] accounting for x, y, z,ẋ,ẏ,ż,ẍ,ÿ,z, respectively. 1) An Ellipsoid: According to (16), an ellipsoid obstacle H is defined by c and A in Fig. 1(b). Given the start condition s s = [3, 3, 3, 3.2, 0, 2, 0, 0, 0] × 100 and the end condition s g , we generate a minimum snap trajectory [23] T1 with 7-th degree polynomials w.r.t. ...
In this letter, we introduce a generalized continuous collision detection (CCD) framework for the mobile robot along the polynomial trajectory in cluttered environments including various static obstacle models. Specifically, we find that the collision conditions between robots and obstacles could be transformed into a set of polynomial inequalities, whose roots can be efficiently solved by the proposed solver. In addition, we test different types of mobile robots with various kinematic and dynamic constraints in our generalized CCD framework and validate that it allows the provable collision checking and can compute the exact time of impact. Furthermore, we combine our architecture with the path planner in the navigation system. Benefiting from our CCD method, the mobile robot is able to work safely in some challenging scenarios.
... Pre-calculation of trajectories for a given task plan provides sufficient flexibility in devising the multiple robotic workstation [48] and enhancing the efficiency in the robotic workcell [49]. Therefore, a common strategy is to pre-calculate a set of trajectories through motion primitives [50,51] or a library of motions [52]. ...
An execution time estimation model is proposed to accurately estimate the execution time in a robotic assembly workcell. The current study exploits a trajectory generation, which is based on cubic splines in particular, because of their natural characteristics, i.e., joint scalability and computational simplicity. It takes into account task specifications and motion capabilities of the robotic arm to estimate the execution time of a task plan. The estimation accuracy is evaluated through a comparison of the computed task times with the results of a simulation of an industrial robot to execute a series of task plans. The evaluation has been done based on the control system (R-30iB Plus) of the Fanuc robot CR-7iA/L in Roboguide software (RG). The results show that the proposed framework is capable of estimating the task time with reasonable accuracy. Possible directions to further improve this accuracy are discussed.
... In other methods, researchers use polynomial splines [16], clothoids [17], and generalized Cornu spirals [18] to obtain smooth curves. Note, that B-splines are most widely used today [19][20][21][22]. Using a cubic B-spline and five control points allows achieving a continuous, smooth change in the path curvature. ...
... hereinafter j = 1, 2, 3. The command signals for the generator (20) are the current values of the reference vector signal χ 1 (t) = (χ 11 (t), χ 12 (t), χ 13 (t)) T , which are assumed to be continuous or piecewise continuous deterministic functions of time. Smoothness requirements are not imposed on them, their analytical description is generally unknown, but it is assumed that their first derivatives are bounded: ...
... Constraints (22) should be understood as one-sided at the points of discontinuity of the first derivative. Similar to Subsection 3.1, for the dynamic model (20), we pose the problem of tracking the reference trajectory χ 1 (t) for the output variables of the generator x 1 (t). It is solved by selecting corrective actions w in the form of feedback from known signals χ 1 (t), x 1 (t), x 2 (t). ...
In the deterministic formulation, two problems are considered related to the generation of admissible reference actions for the automatic control of an autonomous mobile robot in the problem of path following stabilization. The first problem is the restoration of the derivatives of the reference vector signal without its analytical description. This signal sets the realizable trajectory for the robot. An easy-to-tune dynamic differentiator with piecewise linear corrective actions is proposed to solve that problem. The differentiator is constructed as a copy of the virtual canonical model with unknown input and provides an estimation of derivatives of any required order with any given accuracy. The second problem is the smoothing of composite spatial trajectories. For solution, a dynamic generator is presented. It is constructed as a copy of the motion equations of a specific robot. A decomposition procedure for the synthesis of S-shaped smooth and bounded corrective actions of the generator is developed. In the generator, a reference non-smooth vector signal is used to describe the path of the robot in the first approximation. For tuning the generator, the constraints on the state variables and controls of a specific robot are taken into account. It produces smooth output signals, as well as their derivatives, realizable by a particular robot. These algorithms can be implemented both at the planning stage and in real time since they do not require large computational costs. Their effectiveness is verified by numerical simulation for the movement of the center of mass of an unmanned aerial vehicle.
... The linear assumption, as we described in Section 1, meaning all vertices of the meshes move in linear trajectories during each time step, is well established and commonly used. However, rigid motions [TKM09,RKC02,Can86,ZRLK07] and polynomial trajectories [PZM12] also require corresponding CCD algorithms to avoid approximation errors caused by linearization. ...
We introduce the first exact root parity counter for continuous collision detection (CCD). That is, our algorithm computes the parity (even or odd) of the number of roots of the cubic polynomial arising from a CCD query. We note that the parity is unable to differentiate between zero (no collisions) and the rare case of two roots (collisions).
Our method does not have numerical parameters to tune, has a performance comparable to efficient approximate algorithms, and is exact. We test our approach on a large collection of synthetic tests and real simulations, and we demonstrate that it can be easily integrated into existing simulators.
