Simulink Model

The reinforcement learning environment for this example consists in a simple bicycle model for the ego car together with a simple longitudinal model for the lead car. The training goal is to make the ego car travel at a set velocity while maintaining a safe distance from lead car by controlling longitudinal acceleration and braking, and also while keeping the ego car travelling along the centerline of its lane by controlling the front steering angle. For more information on PFC, see Path Following Control System (Model Predictive Control Toolbox). The ego car dynamics are specified by the following parameters.

m = 1600;   % total vehicle mass (kg)
Iz = 2875;  % yaw moment of inertia (mNs^2)
lf = 1.4;   % long. distance from center of gravity to front tires (m)
lr = 1.6;   % long. distance from center of gravity to rear tires (m)
Cf = 19000; % cornering stiffness of front tires (N/rad)
Cr = 33000; % cornering stiffness of rear tires (N/rad)
tau = 0.5;  % longitudinal time constant

Specify the initial position and velocity for the two vehicles.

x0_lead = 50;   % initial position for lead car (m)
v0_lead = 24;   % initial velocity for lead car (m/s)
x0_ego = 10;    % initial position for ego car (m)
v0_ego = 18;    % initial velocity for ego car (m/s)

Specify the standstill default spacing (m), time gap (s), and driver-set velocity (m/s).

D_default = 10;
t_gap = 1.4;
v_set = 28;

To simulate the physical limitations of the vehicle dynamics, constrain the acceleration to the range [–3,2] (m/s^2), and the steering angle to the range [–0.2618,0.2618] (rad), that is -15 and 15 degrees.

amin_ego = -3;
amax_ego = 2;
umin_ego = -0.2618; % +15 deg
umax_ego = 0.2618; % -15 deg

The curvature of the road is defined by a constant 0.001 ($$m^{-1}$$). The initial value for lateral deviation is 0.2 m and the initial value for the relative yaw angle is –0.1 rad.

rho = 0.001;
e1_initial = 0.2;
e2_initial = -0.1;

Define the sample time Ts and simulation duration Tf in seconds.

Ts = 0.1;
Tf = 60;

Open the model.

mdl = "rlPFCMdl";
open_system(mdl)
agentblk = mdl + "/RL Agent";

For this model:

where $$u_{t-1}$$ is the steering input from the previous time step $\mathit{t}-1$, $a_{t-1}$ is the acceleration input from the previous time step. The three logical values are as follows.

The three logical terms in the reward encourage the agent to make both lateral error and velocity error small, and in the meantime, penalize the agent if the simulation is terminated early.

Create Environment Interface

Create an environment interface for the Simulink model.

Create the observation specification.

obsInfo = rlNumericSpec([9 1], ...
    LowerLimit=-inf*ones(9,1), ...
    UpperLimit=inf*ones(9,1));
obsInfo.Name = "observations";

Create the action specification.

actInfo = rlNumericSpec([2 1], ...
    LowerLimit=[-3;-0.2618], ...
    UpperLimit=[2;0.2618]);
actInfo.Name = "accel;steer";

Create the environment interface.

env = rlSimulinkEnv(mdl,agentblk,obsInfo,actInfo);

To define the initial conditions, specify an environment reset function using an anonymous function handle. The reset function localResetFcn, which is defined at the end of the example, randomizes the initial position of the lead car, the lateral deviation, and the relative yaw angle.

env.ResetFcn = @(in)localResetFcn(in);

Fix the random generator seed for reproducibility.

rng(0)

Create DDPG Agent

DDPG agents use a parametrized Q-value function approximator to estimate the value of the policy. A Q-value function critic takes the current observation and an action as inputs and returns a single scalar as output (the estimated discounted cumulative long-term reward given the action from the state corresponding to the current observation, and following the policy thereafter).

To model the parametrized Q-value function within the critic, use a neural network with two input layers (one for the observation channel, as specified by obsInfo, and the other for the action channel, as specified by actInfo) and one output layer (which returns the scalar value). Note that prod(obsInfo.Dimension) and prod(actInfo.Dimension) return the number of dimensions of the observation and action spaces, respectively, regardless of whether they are arranged as row vectors, column vectors, or matrices.

Define each network path as an array of layer objects, and assign names to the input and output layers of each path. These names allow you to connect the paths and then later explicitly associate the network input and output layers with the appropriate environment channel.

% Number of neurons
L = 100;

% Main path
mainPath = [
    featureInputLayer(prod(obsInfo.Dimension),Name="obsInLyr")
    fullyConnectedLayer(L)
    reluLayer
    fullyConnectedLayer(L)
    additionLayer(2,Name="add")
    reluLayer
    fullyConnectedLayer(L)
    reluLayer
    fullyConnectedLayer(1,Name="QValLyr")
    ];

% Action path
actionPath = [
    featureInputLayer(prod(actInfo.Dimension),Name="actInLyr")
    fullyConnectedLayer(L,Name="actOutLyr")
    ];

% Assemble layergraph object
criticNet = layerGraph(mainPath);
criticNet = addLayers(criticNet,actionPath);    
criticNet = connectLayers(criticNet,"actOutLyr","add/in2");

Convert to dlnetwork object and display the number of weights.

criticNet = dlnetwork(criticNet);
summary(criticNet)

   Initialized: true

   Number of learnables: 21.6k

   Inputs:
      1   'obsInLyr'   9 features
      2   'actInLyr'   2 features

View the critic network configuration.

figure
plot(criticNet)

Create the critic using the specified neural network and the environment action and observation specifications. Pass as additional arguments also the names of the network layers to be connected with the observation and action channels. For more information, see rlQValueFunction.

critic = rlQValueFunction(criticNet,obsInfo,actInfo,...
    ObservationInputNames="obsInLyr",ActionInputNames="actInLyr");

DDPG agents use a parametrized deterministic policy over continuous action spaces, which is learned by a continuous deterministic actor. This actor takes the current observation as input and returns as output an action that is a deterministic function of the observation.

To model the parametrized policy within the actor, use a neural network with one input layer (which receives the content of the environment observation channel, as specified by obsInfo) and one output layer (which returns the action to the environment action channel, as specified by actInfo).

Define the network as an array of layer objects.

actorNet = [
    featureInputLayer(prod(obsInfo.Dimension))
    fullyConnectedLayer(L)
    reluLayer
    fullyConnectedLayer(L)
    reluLayer
    fullyConnectedLayer(L)
    reluLayer
    fullyConnectedLayer(2)
    tanhLayer
    scalingLayer(Scale=[2.5;0.2618],Bias=[-0.5;0])
    ];

Convert to dlnetwork object and display the number of weights.

actorNet = dlnetwork(actorNet);
summary(actorNet)

   Initialized: true

   Number of learnables: 21.4k

   Inputs:
      1   'input'   9 features

Construct the actor similarly to the critic. For more information, see rlContinuousDeterministicActor.

actor = rlContinuousDeterministicActor(actorNet,obsInfo,actInfo);

Specify training options for the critic and the actor using rlOptimizerOptions.

criticOptions = rlOptimizerOptions( ...
    LearnRate=1e-3, ...
    GradientThreshold=1, ...
    L2RegularizationFactor=1e-4);
actorOptions = rlOptimizerOptions( ...
    LearnRate=1e-4, ...
    GradientThreshold=1, ...
    L2RegularizationFactor=1e-4);

Specify the DDPG agent options using rlDDPGAgentOptions, include the options for the actor and the critic.

agentOptions = rlDDPGAgentOptions(...
    SampleTime=Ts,...
    ActorOptimizerOptions=actorOptions,...
    CriticOptimizerOptions=criticOptions,...
    ExperienceBufferLength=1e6);
agentOptions.NoiseOptions.Variance = [0.6;0.1];
agentOptions.NoiseOptions.VarianceDecayRate = 1e-5;

Create the DDPG agent using the actor, the critic, and the agent options. For more information, see rlDDPGAgent.

agent = rlDDPGAgent(actor,critic,agentOptions);

Train Agent

To train the agent, first specify the training options. For this example, use the following options:

For more information, see rlTrainingOptions.

maxepisodes = 1e4;
maxsteps = ceil(Tf/Ts);
trainingOpts = rlTrainingOptions(...
    MaxEpisodes=maxepisodes,...
    MaxStepsPerEpisode=maxsteps,...
    Verbose=false,...
    Plots="training-progress",...
    StopTrainingCriteria="EpisodeCount",...
    StopTrainingValue=1450);

Train the agent using the train function. Training is a computationally intensive process that takes several minutes to complete. To save time while running this example, load a pretrained agent by setting doTraining to false. To train the agent yourself, set doTraining to true.

doTraining = false;

if doTraining    
    % Train the agent.
    trainingStats = train(agent,env,trainingOpts);
else
    % Load a pretrained agent for the example.
    load("SimulinkPFCDDPG.mat","agent")       
end

Simulate DDPG Agent

To validate the performance of the trained agent, simulate the agent within the Simulink environment by uncommenting the following commands. For more information on agent simulation, see rlSimulationOptions and sim.

% simOptions = rlSimulationOptions(MaxSteps=maxsteps);
% experience = sim(env,agent,simOptions);

To demonstrate the trained agent using deterministic initial conditions, simulate the model in Simulink.

e1_initial = -0.4;
e2_initial = 0.1;
x0_lead = 80;
sim(mdl)

The following plots show the simulation results when the lead car is 70 (m) ahead of the ego car.

Close the Simulink model.

bdclose(mdl)

Reset Function

function in = localResetFcn(in)
    
    % random value for initial position of lead car
    in = setVariable(in,"x0_lead",40+randi(60,1,1));
    
    % random value for lateral deviation
    in = setVariable(in,"e1_initial", 0.5*(-1+2*rand));
    
    % random value for relative yaw angle
    in = setVariable(in,"e2_initial", 0.1*(-1+2*rand));

end