c# 关节空间五次多项式轨迹规划

根据提供的引用内容，我们可以得知关节空间轨迹规划是将机器人的关节变量变换成时间的函数，然后对角速度和角加速度进行约束。而五次多项式插值法是一种常用的插值方法，可以用于机械臂轨迹规划。因此，我们可以采用五次多项式插值法进行关节空间轨迹规划。

以下是C#实现关节空间五次多项式轨迹规划的示例代码：

using System;

public class JointSpaceTrajectoryPlanning
{
    // 五次多项式插值法
    public static double[] QuinticPolynomial(double q0, double qf, double v0, double vf, double a0, double af, double tf, double t)
    {
        double[] result = new double[3];
        double a = a0;
        double b = v0;
        double c = q0;
        double d = (6 * qf - 6 * q0 - 3 * vf * tf - 3 * v0 * tf - af * Math.Pow(tf, 2) + a0 * Math.Pow(tf, 2)) / Math.Pow(tf, 3);
        double e = (-15 * qf + 15 * q0 + 8 * vf * tf + 7 * v0 * tf + 3 * af * Math.Pow(tf, 2) - 2 * a0 * Math.Pow(tf, 2)) / Math.Pow(tf, 4);
        double f = (10 * qf - 10 * q0 - 6 * vf * tf - 4 * v0 * tf - af * Math.Pow(tf, 2) + a0 * Math.Pow(tf, 2)) / Math.Pow(tf, 5);

        result[0] = a + 2 * b * t + 3 * c * Math.Pow(t, 2) + 4 * d * Math.Pow(t, 3) + 5 * e * Math.Pow(t, 4) + 6 * f * Math.Pow(t, 5);
        result[1] = 2 * a + 6 * b * t + 12 * c * Math.Pow(t, 2) + 20 * d * Math.Pow(t, 3) + 30 * e * Math.Pow(t, 4) + 42 * f * Math.Pow(t, 5);
        result[2] = 6 * b + 24 * c * t + 60 * d * Math.Pow(t, 2) + 120 * e * Math.Pow(t, 3) + 210 * f * Math.Pow(t, 4);

        return result;
    }

    // 关节空间轨迹规划
    public static double[,] JointSpaceTrajectory(double[] q0, double[] qf, double[] v0, double[] vf, double[] a0, double[] af, double tf, double dt)
    {
        int n = (int)Math.Ceiling(tf / dt) + 1;
        double[,] result = new double[n, q0.Length];

        for (int i = 0; i < q0.Length; i++)
        {
            double[] q = QuinticPolynomial(q0[i], qf[i], v0[i], vf[i], a0[i], af[i], tf, 0);
            double[] v = QuinticPolynomial(q0[i], qf[i], v0[i], vf[i], a0[i], af[i], tf, dt);
            double[] a = QuinticPolynomial(q0[i], qf[i], v0[i], vf[i], a0[i], af[i], tf, 2 * dt);

            result[0, i] = q[0];
            result[1, i] = v[0];
            result[2, i] = a[0];

            for (int j = 1; j < n; j++)
            {
                double t = j * dt;
                q = QuinticPolynomial(q0[i], qf[i], v0[i], vf[i], a0[i], af[i], tf, t);
                v = QuinticPolynomial(q0[i], qf[i], v0[i], vf[i], a0[i], af[i], tf, t + dt);
                a = QuinticPolynomial(q0[i], qf[i], v0[i], vf[i], a0[i], af[i], tf, t + 2 * dt);

                result[j, i] = q[0];
            }
        }

        return result;
    }
}

// 示例
public class Example
{
    public static void Main()
    {
        double[] q0 = { 0, 0, 0 };
        double[] qf = { 1, 1, 1 };
        double[] v0 = { 0, 0, 0 };
        double[] vf = { 0, 0, 0 };
        double[] a0 = { 0, 0, 0 };
        double[] af = { 0, 0, 0 };
        double tf = 1;
        double dt = 0.01;

        double[,] result = JointSpaceTrajectoryPlanning.JointSpaceTrajectory(q0, qf, v0, vf, a0, af, tf, dt);

        for (int i = 0; i < result.GetLength(0); i++)
        {
            for (int j = 0; j < result.GetLength(1); j++)
            {
                Console.Write(result[i, j] + " ");
            }
            Console.WriteLine();
        }
    }
}