In the classic control scheme of the autonomous driving industry, the solution of lateral control and longitudinal control is an independent algorithm for model decoupling. Although this "horizontal and vertical separation" control scheme is feasible, it obviously does not conform to the human driving style, nor does it conform to the objective fact that the horizontal and vertical are closely connected. This paper introduces an implementation scheme of integrated horizontal and vertical unmanned vehicle control, which is more advantageous in describing the vehicle horizontal and vertical coupling, considering the horizontal and vertical joint constraints, and coordinating the horizontal and vertical tracking performance.
01 Unmanned vehicles under the control of two "drivers"
Can a car be driven by two drivers at the same time? It seems that no car is driven like this except the coach car of the driving school.
But this question is a bit weird in the eyes of ordinary people, and its answer is affirmative for the field of unmanned driving-the classic vehicle control scheme is realized through the cooperation of two controllers that control the vertical and horizontal respectively.
As shown in Figure 1, in the classic control scheme of the autopilot industry, the solution of lateral control and longitudinal control is an independent algorithm of model decoupling [1,2], which cooperates only through one-way, single-pass dependent parameter transfer ( The dashed box is the control module, and the arrow indicates the parameter dependency).
It is not difficult to imagine that this is equivalent to "two drivers driving the same car". The vertical driver operates the pedal (no need to look at the road, only the speedometer) to achieve the planned speed, and the horizontal driver operates the steering wheel (need to look at the road) to achieve the plan. path. The control module combines the power of two people to "realize" the planned trajectory (including the planned path and the planned speed) given by the upstream planning module as accurately as possible.
At first glance, this “unreasonable” horizontal and vertical separation control scheme is widely adopted by the industry for its reasons:
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Smaller issues-decompose the trajectory tracking problem into two vertical and horizontal sub-problems, which can reduce the complexity of a single problem, and avoid non-linearities such as cross-products in the model (thereby adopting a linear horizontal model), which is conducive to control Solve quickly.
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The coupling is not strong-the steering structure of conventional vehicles is often limited to an angle of 30 degrees, the curvature of the driving path is limited, the component of the steering wheel lateral force on the longitudinal axis of the vehicle body is small, and the longitudinal control is basically not affected by the lateral state and can be performed independently.
02 Potential problems of horizontal and vertical separation
Although the classic horizontal and vertical separation control scheme is feasible, it obviously does not conform to human driving style.
In fact, for the task of trajectory tracking, horizontal and vertical are closely related [3-5], mainly including the following three aspects.
(1) Vehicle dynamic modeling includes horizontal and vertical coupling
There is a natural horizontal and vertical coupling in real vehicles, and real old drivers will consider the mutual influence between the horizontal and vertical behaviors of the vehicle when controlling. For example, the longitudinal vehicle speed when cornering will affect the lateral acceleration, and the steering action will also have a longitudinal braking effect. In fact, kinematic coupling [1], tire force coupling [2], and load transfer coupling [3] are the three major transverse and longitudinal couplings in the vehicle model [4].
In the horizontal and vertical separation control scheme, the vertical control and the horizontal control use independent models, and can only interact through state parameters. Therefore, the above coupling cannot be described reasonably before the solution, and the accuracy of the model will further affect the control solution. Optimality.
(2) The construction of driving constraints requires the participation of both horizontal and vertical
The driving constraint determines the feasible space of the control amount, which can be understood as the old driver's grasp of the driving safety range and vehicle performance limit. There are some driving constraints descriptions that can be based on the characteristics of a single horizontal/vertical direction, such as upper steering angle constraints, pedal upper limits, etc.; but there are also some driving constraints that require the participation of both horizontal and vertical characteristics, such as location boundaries. Constraints (the abscissa and the ordinate participate together), the maximum centripetal acceleration constraint (the lateral rotation angle and the longitudinal velocity participate together), etc.
In the horizontal and vertical separation control scheme, the horizontal and vertical control algorithms each only have a solution space in one direction, which cannot describe the driving constraints of the horizontal and vertical joint participation. Taking the maximum centripetal acceleration constraint as an example, the lateral control in the separation scheme can only prevent the centripetal acceleration from exceeding the limit by restraining the steering angle and not decelerating. In fact, a part of the feasible set is lost, as shown in the following figure:
(3) There is horizontal and vertical competition in tracking performance evaluation
Trajectory tracking focuses on the comprehensive tracking performance indicators in both horizontal and vertical directions, and the two unidirectional indicators have a certain competitive relationship [6]. In the horizontal and vertical separation control scheme, the horizontal and vertical control algorithms are independent, and they only focus on tracking performance indicators in one direction. In some scenarios, the horizontal and vertical tracking performance may be unbalanced.
For example, when cornering at high speed, the longitudinal control can easily track the target speed, but it will improve the accuracy requirements of the lateral control in time and angle, and increase the lateral error. But from the perspective of overall planning, the vertical tracking at this time should give way and actively slow down, in exchange for more performance improvements in the horizontal direction.
03 The design idea of the horizontal and vertical integrated scheme
Aiming at the above-mentioned problems of horizontal and vertical separation control, we try to develop an engineering feasible horizontal and vertical integrated control scheme.
In view of the above-mentioned three connections between the horizontal and vertical in the trajectory tracking task, the design points of the horizontal and vertical integrated control scheme include:
(1) Vehicle modeling using horizontal and vertical coupling —then describe the dynamic characteristics of the vehicle more accurately.
(2) Adopt the constraint form of horizontal and vertical joint -and then construct the feasible set of control variables more complete.
(3) Use horizontal and vertical comprehensive performance evaluation -and then coordinate the two one-way tracking performance.
It should be noted that because it is difficult to avoid non-linearities such as cross-products in the horizontal and vertical coupling modeling [1,7], overcoming the non-linear problem and making the controller reach the practical calculation speed of engineering is the core difficulty in the realization of the scheme .
In the existing literature, the control research considering the horizontal and vertical coupling model usually adopts a scheme based on sliding mode control (SMC) [3,4,8-10], constructing a specific variable structure control law to directly perform nonlinear control, Keep the evolution of the system state within the pre-designed sliding surface neighborhood (the sliding surface is usually designed as a state surface where the horizontal and vertical errors gradually reach zero). However, the nonlinear control law cannot consider the state constraints, and its construction result and difficulty are strongly related to the model, and the transferability of the control law is poor.
Here, we adopt another aspect - a linear time-varying model predictive control (Linear Time-Varying Model Predictive Control , LTV-MPC) for the vertical and horizontal nonlinear coupled model.
The control effect and calculation performance of this scheme are between conventional (linear and time-invariant) MPC and nonlinear MPC. It is a practical control method for nonlinear or time-varying systems [11-13], and has flexible Constraint handling capacity [1,13].
The specific implementation of the LTV-MPC solution will be introduced in the next part.
In summary, the essence of the scheme is to construct an optimization problem of a control variable sequence in each control frame. The three design points of horizontal and vertical coupling modeling, horizontal and vertical joint constraints, and horizontal and vertical tracking are corresponding to this optimization problem. State equation constraints, inequality constraints, and evaluation functions are shown in Figure 3.
It should be noted that the "linear" in "linear time-varying model predictive control" means that the state equation constraints and inequality constraints entering the solver are linear in every frame to ensure the calculation speed; while "time-varying" means Between different frames, linear state equation constraints and inequality constraints will change to describe the nonlinearity of the real system.
04 The concrete realization of LTV-MPC horizontal and vertical integrated control
4.1 Construction of horizontal and vertical coupling dynamics and combined horizontal and vertical constraints
The modeling process needs to fully describe all or the main horizontal and vertical coupling of the controlled vehicle, the horizontal and vertical control variables (such as lateral steering angle, longitudinal acceleration) u and the horizontal and vertical state variables (such as vehicle position, yaw angle, longitudinal speed) x is added to the model in the form of variables to be decided, and the horizontal and vertical joint constraints that are more in line with the real-world scenario are constructed.
The horizontal and vertical coupled two-wheel kinematics model shown in the following figure is taken as an example. Take the system state quantity as and the control quantity as
(the acceleration command α will be mapped to the pedal command through a table lookup later), and its dynamic characteristics can be modeled as:
Taking the maximum centripetal acceleration as an example, the horizontal and vertical joint constraints of the system can be modeled as:
Although the two-wheel kinematics model is simple, it has described the most important horizontal and vertical coupling of the vehicle-kinematic coupling. If a more complex vehicle model is used, the horizontal and vertical coupling modeling can also be completed in a similar way.
4.2 Linearization of dynamic characteristics and constraints
The horizontal and vertical coupling model obtained in the previous step usually contains nonlinear terms such as the cross product of variables and trigonometric functions, which need to be linearized to reduce the complexity-this is the core step to deal with the nonlinear difficulty of the horizontal and vertical coupling model. The core idea is to select the current state quantity and the previous frame control quantity as the "base point", and then perform the first-order Taylor expansion of the nonlinear model at the base point [4].
The rationality of this method is that within a short period of prediction domain (such as 1 second) in the future, the system state and control quantity will basically change near the current time value , so the linearized model obtained at this point basically has enough Describe accuracy.
Specifically, the linearization results of the dynamic characteristics are:
among them:
Still taking the two-wheel kinematics model as an example, there are:
The linearization result of the horizontal and vertical joint constraint is:
among them:
Still taking the maximum centripetal acceleration as an example, there are:
After linearization is completed, in order to adapt the continuous model to the numerical calculation solver, it is necessary to adopt a specific discretization method according to a specific discrete time interval (usually the length of a control frame), such as the zero-order discrete matrix transformation formula [14] :
Perform discretization as accurately as possible to obtain a discrete linear model:
The subscript represents the value of the vector to be solved at the time, and t is the current time.
4.3 Construction of the horizontal and vertical comprehensive evaluation function
In order to solve the optimal control quantity, it is necessary to construct an evaluation function of the system state . It should be noted that, compared to the optimization of the tracking performance in a single direction for the horizontal and vertical separation control, the goal of the horizontal and vertical integrated control is the coordination and coordination of the tracking performance in the horizontal and vertical directions. Therefore
, the comprehensive tracking in the horizontal and vertical directions should be described. performance.
Still taking the two-wheel kinematics model as an example, you can define:
Wherein ,
,
,
respectively, velocity error, the longitudinal position error, lateral position error, the yaw angle error in the total evaluation function of the weight penalty - two front portions of both the longitudinal evaluation, the evaluation of the lateral portion. The superscript
indicates the planned trajectory, which is given by the upstream planning module.
Under such evaluation function design, the smaller the value at a certain moment , the better the comprehensive tracking performance at that moment. And it is not difficult to see that
the marginal contribution of improving the large error term is greater, so the minimization
can realize the overall coordination of the horizontal and vertical tracking performance, and avoid excessive errors in a certain direction.
For example, in a high-speed cornering scene, the horizontal and vertical integrated controller will perform appropriate active deceleration at the corner, sacrificing a little longitudinal tracking performance in exchange for more lateral tracking performance, making it as small as possible-this overall idea In fact, it is more in line with real motion control needs.
4.4 Solving and issuing control variables
Selecting the number of prediction steps as (prediction domain is
), combining the linearization model obtained in step 4.2 and the evaluation function obtained in step 4.3, the following MPC problem can be formed [5]:
The problem is essentially a linear constraint quadratic programming problem. Next, only need to solve the same as the conventional MPC, the optimal control sequence can be solved , and the state prediction trajectory is
accompanied by it. This trajectory can be visually expressed as a trajectory similar to the planned trajectory and containing speed information.
After it is issued, you can wait for the arrival of the next control frame, and then return to step 4.2 (if you need to update the model, you should return to step 4.1), and execute it in a loop.
05 Summary
Aiming at the inadequacy of the automatic driving horizontal and vertical separation control scheme, which can not describe the vehicle horizontal and vertical coupling, cannot consider the horizontal and vertical joint constraints, and has no horizontal and vertical coordinating ability, a horizontal and vertical integrated vehicle control scheme considering the horizontal and vertical joint constraints is given.
This solution supports the consideration of the vehicle's horizontal and vertical coupling during modeling to improve the accuracy of the model ; the time-varying linear MPC framework is used to overcome the nonlinearity caused by the horizontal and vertical coupling and constraints, and to improve the time-consuming and reliability of the solution ; The optimization of comprehensive tracking performance in the vertical joint feasible region has a certain horizontal and vertical overall planning ability .
Comment
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The horizontal and vertical are defined by the body. When the body rotates, the horizontal and vertical physical quantities at adjacent moments will contribute to each other.
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When the longitudinal speed is high, the lateral force will be more sensitive to steering; when the lateral steering angle is large, the longitudinal force will have an additional braking component.
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When longitudinally decelerating, part of the load is transferred to the front wheels to increase the lateral force of the tire.
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Another commonly used linearization method is to expand at (0,0), but its description accuracy is relatively poor. There is also a linearization method that takes
a base point for each prediction moment
and obtains a cluster of base points [12], which can also achieve better description accuracy, but to simplify the expression, this article will not repeat it.
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Here, it is
written as the matrix form in the objective function
.
references
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[11] BamiehB, Giarre L. Identification of linear parameter varying models[J].International Journal of Robust and Nonlinear Control:IFAC‐Affiliated Journal, 2002, 12(9): 841-853.
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[13] Xing X, Lin J, Brandon N, et al. Time-varying model predictivecontrol of a reversible-SOC energy-storage plant based on the linearparameter-varying method[J]. IEEE Transactions on Sustainable Energy,2020, 11(3): 1589-1600.
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