feat: add simple inverse-kinematics solver for multibodies (#632)

* feat: add a simple jacobian-based inverse-kinematics implementation for multibodies

* feat: add 2d inverse kinematics example

* feat: make forward_kinematics auto-fix the root’s degrees of freedom

* feat: add 3d inverse kinematics example

* chore: update changelog

* chore: clippy fixes

* chore: more clippy fixes

* fix tests

src_testbed/mouse.rs

@@ -0,0 +1,46 @@
+use crate::math::Point;
+use bevy::prelude::*;
+use bevy::window::PrimaryWindow;
+
+#[derive(Component)]
+pub struct MainCamera;
+
+#[derive(Default, Copy, Clone, Debug, Resource)]
+pub struct SceneMouse {
+    #[cfg(feature = "dim2")]
+    pub point: Option<Point<f32>>,
+    #[cfg(feature = "dim3")]
+    pub ray: Option<(Point<f32>, crate::math::Vector<f32>)>,
+}
+
+pub fn track_mouse_state(
+    mut scene_mouse: ResMut<SceneMouse>,
+    windows: Query<&Window, With<PrimaryWindow>>,
+    camera: Query<(&GlobalTransform, &Camera), With<MainCamera>>,
+) {
+    if let Ok(window) = windows.get_single() {
+        for (camera_transform, camera) in camera.iter() {
+            if let Some(cursor) = window.cursor_position() {
+                let ndc_cursor = ((cursor / Vec2::new(window.width(), window.height()) * 2.0)
+                    - Vec2::ONE)
+                    * Vec2::new(1.0, -1.0);
+                let ndc_to_world =
+                    camera_transform.compute_matrix() * camera.projection_matrix().inverse();
+                let ray_pt1 =
+                    ndc_to_world.project_point3(Vec3::new(ndc_cursor.x, ndc_cursor.y, -1.0));
+
+                #[cfg(feature = "dim2")]
+                {
+                    scene_mouse.point = Some(Point::new(ray_pt1.x, ray_pt1.y));
+                }
+                #[cfg(feature = "dim3")]
+                {
+                    let ray_pt2 =
+                        ndc_to_world.project_point3(Vec3::new(ndc_cursor.x, ndc_cursor.y, 1.0));
+                    let ray_dir = ray_pt2 - ray_pt1;
+                    scene_mouse.ray = Some((na::Vector3::from(ray_pt1).into(), ray_dir.into()));
+                }
+            }
+        }
+    }
+}